Dense matrices using a NumPy backend.
This serves as a base class for dense matrices over Real Double Field and Complex Double Field.
AUTHORS:
EXAMPLES:
sage: b=Mat(RDF,2,3).basis()
sage: b[0]
[1.0 0.0 0.0]
[0.0 0.0 0.0]
We deal with the case of zero rows or zero columns:
sage: m = MatrixSpace(RDF,0,3)
sage: m.zero_matrix()
[]
TESTS:
sage: a = matrix(RDF,2,range(4), sparse=False)
sage: TestSuite(a).run()
sage: a = matrix(CDF,2,range(4), sparse=False)
sage: TestSuite(a).run()
Bases: sage.matrix.matrix_dense.Matrix_dense
Base class for matrices over the Real Double Field and the Complex Double Field. These are supposed to be fast matrix operations using C doubles. Most operations are implemented using numpy which will call the underlying BLAS on the system.
This class cannot be instantiated on its own. The numpy matrix creation depends on several variables that are set in the subclasses.
EXAMPLES:
sage: m = Matrix(RDF, [[1,2],[3,4]])
sage: m**2
[ 7.0 10.0]
[15.0 22.0]
sage: n= m^(-1); n
[-2.0 1.0]
[ 1.5 -0.5]
Returns a decomposition of the (row-permuted) matrix as a product of a lower-triangular matrix (“L”) and an upper-triangular matrix (“U”).
OUTPUT:
For an \(m\times n\) matrix A this method returns a triple of immutable matrices P, L, U such that
The computed decomposition is cached and returned on subsequent calls, thus requiring the results to be immutable.
Effectively, P permutes the rows of A. Then L can be viewed as a sequence of row operations on this matrix, where each operation is adding a multiple of a row to a subsequent row. There is no scaling (thus 1’s on the diagonal of L) and no row-swapping (P does that). As a result U is close to being the result of Gaussian-elimination. However, round-off errors can make it hard to determine the zero entries of U.
Note
Sometimes this decomposition is written as A=P*L*U, where P represents the inverse permutation and is the matrix inverse of the P returned by this method. The computation of this matrix inverse can be accomplished quickly with just a transpose as the matrix is orthogonal/unitary.
EXAMPLES:
sage: m = matrix(RDF,4,range(16))
sage: P,L,U = m.LU()
sage: P*m
[12.0 13.0 14.0 15.0]
[ 0.0 1.0 2.0 3.0]
[ 8.0 9.0 10.0 11.0]
[ 4.0 5.0 6.0 7.0]
sage: L*U
[12.0 13.0 14.0 15.0]
[ 0.0 1.0 2.0 3.0]
[ 8.0 9.0 10.0 11.0]
[ 4.0 5.0 6.0 7.0]
Trac 10839 made this routine available for rectangular matrices.
sage: A = matrix(RDF, 5, 6, range(30)); A
[ 0.0 1.0 2.0 3.0 4.0 5.0]
[ 6.0 7.0 8.0 9.0 10.0 11.0]
[12.0 13.0 14.0 15.0 16.0 17.0]
[18.0 19.0 20.0 21.0 22.0 23.0]
[24.0 25.0 26.0 27.0 28.0 29.0]
sage: P, L, U = A.LU()
sage: P
[0.0 0.0 0.0 0.0 1.0]
[1.0 0.0 0.0 0.0 0.0]
[0.0 0.0 1.0 0.0 0.0]
[0.0 0.0 0.0 1.0 0.0]
[0.0 1.0 0.0 0.0 0.0]
sage: L.zero_at(0) # Use zero_at(0) to get rid of signed zeros
[ 1.0 0.0 0.0 0.0 0.0]
[ 0.0 1.0 0.0 0.0 0.0]
[ 0.5 0.5 1.0 0.0 0.0]
[0.75 0.25 0.0 1.0 0.0]
[0.25 0.75 0.0 0.0 1.0]
sage: U.zero_at(0) # Use zero_at(0) to get rid of signed zeros
[24.0 25.0 26.0 27.0 28.0 29.0]
[ 0.0 1.0 2.0 3.0 4.0 5.0]
[ 0.0 0.0 0.0 0.0 0.0 0.0]
[ 0.0 0.0 0.0 0.0 0.0 0.0]
[ 0.0 0.0 0.0 0.0 0.0 0.0]
sage: P*A-L*U
[0.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0 0.0]
sage: P.transpose()*L*U
[ 0.0 1.0 2.0 3.0 4.0 5.0]
[ 6.0 7.0 8.0 9.0 10.0 11.0]
[12.0 13.0 14.0 15.0 16.0 17.0]
[18.0 19.0 20.0 21.0 22.0 23.0]
[24.0 25.0 26.0 27.0 28.0 29.0]
Trivial cases return matrices of the right size and characteristics.
sage: A = matrix(RDF, 5, 0, entries=0)
sage: P, L, U = A.LU()
sage: P.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
sage: L.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
sage: U.parent()
Full MatrixSpace of 5 by 0 dense matrices over Real Double Field
sage: P*A-L*U
[]
The results are immutable since they are cached.
sage: P, L, U = matrix(RDF, 2, 2, range(4)).LU()
sage: L[0,0] = 0
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
sage: P[0,0] = 0
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
sage: U[0,0] = 0
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
Returns True if the LU form of this matrix has already been computed.
EXAMPLES:
sage: A = random_matrix(RDF,3) ; A.LU_valid()
False
sage: P, L, U = A.LU()
sage: A.LU_valid()
True
Returns a factorization into a unitary matrix and an upper-triangular matrix.
INPUT:
Any matrix over RDF or CDF.
OUTPUT:
Q, R – a pair of matrices such that if \(A\) is the original matrix, then
where \(R\) is upper-triangular. \(Q^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. So \(Q\) is a unitary matrix (or rather, orthogonal, in the real case), or equivalently \(Q\) has orthogonal columns. For a matrix of full rank this factorization is unique up to adjustments via multiples of rows and columns by multiples with scalars having modulus \(1\). So in the full-rank case, \(R\) is unique if the diagonal entries are required to be positive real numbers.
The resulting decomposition is cached.
ALGORITHM:
Calls “linalg.qr” from SciPy, which is in turn an interface to LAPACK routines.
EXAMPLES:
Over the reals, the inverse of Q is its transpose, since including a conjugate has no effect. In the real case, we say Q is orthogonal.
sage: A = matrix(RDF, [[-2, 0, -4, -1, -1],
... [-2, 1, -6, -3, -1],
... [1, 1, 7, 4, 5],
... [3, 0, 8, 3, 3],
... [-1, 1, -6, -6, 5]])
sage: Q, R = A.QR()
At this point, Q is only well-defined up to the signs of its columns, and similarly for R and its rows, so we normalize them:
sage: Qnorm = Q._normalize_columns()
sage: Rnorm = R._normalize_rows()
sage: Qnorm.round(6).zero_at(10^-6)
[ 0.458831 0.126051 0.381212 0.394574 0.68744]
[ 0.458831 -0.47269 -0.051983 -0.717294 0.220963]
[-0.229416 -0.661766 0.661923 0.180872 -0.196411]
[-0.688247 -0.189076 -0.204468 -0.09663 0.662889]
[ 0.229416 -0.535715 -0.609939 0.536422 -0.024551]
sage: Rnorm.round(6).zero_at(10^-6)
[ 4.358899 -0.458831 13.076697 6.194225 2.982405]
[ 0.0 1.670172 0.598741 -1.29202 6.207997]
[ 0.0 0.0 5.444402 5.468661 -0.682716]
[ 0.0 0.0 0.0 1.027626 -3.6193]
[ 0.0 0.0 0.0 0.0 0.024551]
sage: (Q*Q.transpose()).zero_at(10^-14)
[1.0 0.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0 0.0]
[0.0 0.0 1.0 0.0 0.0]
[0.0 0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 0.0 1.0]
sage: (Q*R - A).zero_at(10^-14)
[0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0 0.0]
Now over the complex numbers, demonstrating that the SciPy libraries are (properly) using the Hermitian inner product, so that Q is a unitary matrix (its inverse is the conjugate-transpose).
sage: A = matrix(CDF, [[-8, 4*I + 1, -I + 2, 2*I + 1],
... [1, -2*I - 1, -I + 3, -I + 1],
... [I + 7, 2*I + 1, -2*I + 7, -I + 1],
... [I + 2, 0, I + 12, -1]])
sage: Q, R = A.QR()
sage: Q._normalize_columns().round(6).zero_at(10^-6)
[ 0.730297 0.207057 + 0.538347*I 0.246305 - 0.076446*I 0.238162 - 0.10366*I]
[ -0.091287 -0.207057 - 0.377878*I 0.378656 - 0.195222*I 0.701244 - 0.364371*I]
[ -0.63901 - 0.091287*I 0.170822 + 0.667758*I -0.034115 + 0.040902*I 0.314017 - 0.082519*I]
[-0.182574 - 0.091287*I -0.036235 + 0.07247*I 0.863228 + 0.063228*I -0.449969 - 0.011612*I]
sage: R._normalize_rows().round(6).zero_at(10^-6)
[ 10.954451 -1.917029*I 5.385938 - 2.19089*I -0.273861 - 2.19089*I]
[ 0.0 4.829596 -0.869638 - 5.864879*I 0.993872 - 0.305409*I]
[ 0.0 0.0 12.001608 -0.270953 + 0.442063*I]
[ 0.0 0.0 0.0 1.942964]
sage: (Q.conjugate().transpose()*Q).zero_at(10^-15)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: (Q*R - A).zero_at(10^-14)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
An example of a rectangular matrix that is also rank-deficient. If you run this example yourself, you may see a very small, nonzero entries in the third row, in the third column, even though the exact version of the matrix has rank 2. The final two columns of Q span the left kernel of A (as evidenced by the two zero rows of R). Different platforms will compute different bases for this left kernel, so we do not exhibit the actual matrix.
sage: Arat = matrix(QQ, [[2, -3, 3],
... [-1, 1, -1],
... [-1, 3, -3],
... [-5, 1, -1]])
sage: Arat.rank()
2
sage: A = Arat.change_ring(CDF)
sage: Q, R = A.QR()
sage: R._normalize_rows().round(6).zero_at(10^-6)
[ 5.567764 -2.69408 2.69408]
[ 0.0 3.569585 -3.569585]
[ 0.0 0.0 0.0]
[ 0.0 0.0 0.0]
sage: (Q.conjugate_transpose()*Q).zero_at(10^-14)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
Results are cached, meaning they are immutable matrices. Make a copy if you need to manipulate a result.
sage: A = random_matrix(CDF, 2, 2)
sage: Q, R = A.QR()
sage: Q.is_mutable()
False
sage: R.is_mutable()
False
sage: Q[0,0] = 0
Traceback (most recent call last):
...
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
sage: Qcopy = copy(Q)
sage: Qcopy[0,0] = 679
sage: Qcopy[0,0]
679.0
TESTS:
Trivial cases return trivial results of the correct size, and we check Q itself in one case, verifying a fix for trac ticket #10795.
sage: A = zero_matrix(RDF, 0, 10)
sage: Q, R = A.QR()
sage: Q.nrows(), Q.ncols()
(0, 0)
sage: R.nrows(), R.ncols()
(0, 10)
sage: A = zero_matrix(RDF, 3, 0)
sage: Q, R = A.QR()
sage: Q.nrows(), Q.ncols()
(3, 3)
sage: R.nrows(), R.ncols()
(3, 0)
sage: Q
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
Return the singular value decomposition of this matrix.
The U and V matrices are not unique and may be returned with different values in the future or on different systems. The S matrix is unique and contains the singular values in descending order.
The computed decomposition is cached and returned on subsequent calls.
INPUT:
OUTPUT:
Note that if self is m-by-n, then the dimensions of the matrices that this returns are (m,m), (m,n), and (n, n).
Note
If all you need is the singular values of the matrix, see the more convenient singular_values().
EXAMPLES:
sage: m = matrix(RDF,4,range(1,17))
sage: U,S,V = m.SVD()
sage: U*S*V.transpose()
[ 1.0 2.0 3.0 4.0]
[ 5.0 6.0 7.0 8.0]
[ 9.0 10.0 11.0 12.0]
[13.0 14.0 15.0 16.0]
A non-square example:
sage: m = matrix(RDF, 2, range(1,7)); m
[1.0 2.0 3.0]
[4.0 5.0 6.0]
sage: U, S, V = m.SVD()
sage: U*S*V.transpose()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
S contains the singular values:
sage: S.round(4)
[ 9.508 0.0 0.0]
[ 0.0 0.7729 0.0]
sage: [round(sqrt(abs(x)),4) for x in (S*S.transpose()).eigenvalues()]
[9.508, 0.7729]
U and V are orthogonal matrices:
sage: U # random, SVD is not unique
[-0.386317703119 -0.922365780077]
[-0.922365780077 0.386317703119]
[-0.274721127897 -0.961523947641]
[-0.961523947641 0.274721127897]
sage: (U*U.transpose()).zero_at(1e-15)
[1.0 0.0]
[0.0 1.0]
sage: V # random, SVD is not unique
[-0.428667133549 0.805963908589 0.408248290464]
[-0.566306918848 0.112382414097 -0.816496580928]
[-0.703946704147 -0.581199080396 0.408248290464]
sage: (V*V.transpose()).zero_at(1e-15)
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
TESTS:
sage: m = matrix(RDF,3,2,range(1, 7)); m
[1.0 2.0]
[3.0 4.0]
[5.0 6.0]
sage: U,S,V = m.SVD()
sage: U*S*V.transpose()
[1.0 2.0]
[3.0 4.0]
[5.0 6.0]
sage: m = matrix(RDF, 3, 0, []); m
[]
sage: m.SVD()
([], [], [])
sage: m = matrix(RDF, 0, 3, []); m
[]
sage: m.SVD()
([], [], [])
sage: def shape(x): return (x.nrows(), x.ncols())
sage: m = matrix(RDF, 2, 3, range(6))
sage: map(shape, m.SVD())
[(2, 2), (2, 3), (3, 3)]
sage: for x in m.SVD(): x.is_immutable()
True
True
True
Returns the Cholesky factorization of a matrix that is real symmetric, or complex Hermitian.
INPUT:
Any square matrix with entries from RDF that is symmetric, or with entries from CDF that is Hermitian. The matrix must be positive definite for the Cholesky decomposition to exist.
OUTPUT:
For a matrix \(A\) the routine returns a lower triangular matrix \(L\) such that,
where \(L^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. If the matrix fails to be positive definite (perhaps because it is not symmetric or Hermitian), then this function raises a ValueError.
IMPLEMENTATION:
The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the is_positive_definite() method compute and cache both the Cholesky decomposition and the positive-definiteness. So the is_positive_definite() method or catching a ValueError from the cholesky() method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.
EXAMPLES:
A real matrix that is symmetric and positive definite.
sage: M = matrix(RDF,[[ 1, 1, 1, 1, 1],
... [ 1, 5, 31, 121, 341],
... [ 1, 31, 341, 1555, 4681],
... [ 1,121, 1555, 7381, 22621],
... [ 1,341, 4681, 22621, 69905]])
sage: M.is_symmetric()
True
sage: L = M.cholesky()
sage: L.round(6).zero_at(10^-10)
[ 1.0 0.0 0.0 0.0 0.0]
[ 1.0 2.0 0.0 0.0 0.0]
[ 1.0 15.0 10.723805 0.0 0.0]
[ 1.0 60.0 60.985814 7.792973 0.0]
[ 1.0 170.0 198.623524 39.366567 1.7231]
sage: (L*L.transpose()).round(6).zero_at(10^-10)
[ 1.0 1.0 1.0 1.0 1.0]
[ 1.0 5.0 31.0 121.0 341.0]
[ 1.0 31.0 341.0 1555.0 4681.0]
[ 1.0 121.0 1555.0 7381.0 22621.0]
[ 1.0 341.0 4681.0 22621.0 69905.0]
A complex matrix that is Hermitian and positive definite.
sage: A = matrix(CDF, [[ 23, 17*I + 3, 24*I + 25, 21*I],
... [ -17*I + 3, 38, -69*I + 89, 7*I + 15],
... [-24*I + 25, 69*I + 89, 976, 24*I + 6],
... [ -21*I, -7*I + 15, -24*I + 6, 28]])
sage: A.is_hermitian()
True
sage: L = A.cholesky()
sage: L.round(6).zero_at(10^-10)
[ 4.795832 0.0 0.0 0.0]
[ 0.625543 - 3.544745*I 5.004346 0.0 0.0]
[ 5.21286 - 5.004346*I 13.588189 + 10.721116*I 24.984023 0.0]
[ -4.378803*I -0.104257 - 0.851434*I -0.21486 + 0.371348*I 2.811799]
sage: (L*L.conjugate_transpose()).round(6).zero_at(10^-10)
[ 23.0 3.0 + 17.0*I 25.0 + 24.0*I 21.0*I]
[ 3.0 - 17.0*I 38.0 89.0 - 69.0*I 15.0 + 7.0*I]
[25.0 - 24.0*I 89.0 + 69.0*I 976.0 6.0 + 24.0*I]
[ -21.0*I 15.0 - 7.0*I 6.0 - 24.0*I 28.0]
This routine will recognize when the input matrix is not positive definite. The negative eigenvalues are an equivalent indicator. (Eigenvalues of a Hermitian matrix must be real, so there is no loss in ignoring the imprecise imaginary parts).
sage: A = matrix(RDF, [[ 3, -6, 9, 6, -9],
... [-6, 11, -16, -11, 17],
... [ 9, -16, 28, 16, -40],
... [ 6, -11, 16, 9, -19],
... [-9, 17, -40, -19, 68]])
sage: A.is_symmetric()
True
sage: A.eigenvalues()
[108.07..., 13.02..., -0.02..., -0.70..., -1.37...]
sage: A.cholesky()
Traceback (most recent call last):
...
ValueError: matrix is not positive definite
sage: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I],
... [4 + 2*I, 8, 10*I],
... [2 - 2*I, -10*I, -3]])
sage: B.is_hermitian()
True
sage: [ev.real() for ev in B.eigenvalues()]
[15.88..., 0.08..., -8.97...]
sage: B.cholesky()
Traceback (most recent call last):
...
ValueError: matrix is not positive definite
TESTS:
A trivial case.
sage: A = matrix(RDF, 0, [])
sage: A.cholesky()
[]
The Cholesky factorization is only defined for square matrices.
sage: A = matrix(RDF, 4, 5, range(20))
sage: A.cholesky()
Traceback (most recent call last):
...
ValueError: Cholesky decomposition requires a square matrix, not a 4 x 5 matrix
Returns the condition number of a square nonsingular matrix.
Roughly speaking, this is a measure of how sensitive the matrix is to round-off errors in numerical computations. The minimum possible value is 1.0, and larger numbers indicate greater sensitivity.
INPUT:
OUTPUT:
The condition number of a matrix is the product of a norm of the matrix times the norm of the inverse of the matrix. This requires that the matrix be square and invertible (nonsingular, full rank).
Returned value is a double precision floating point value in RDF, or Infinity. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.
p = 'frob': the default norm employed in computing the condition number, the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes
p = 'sv': the quotient of the maximal and minimal singular value.
p = Infinity or p = oo: the maximum row sum.
p = -Infinity or p = -oo: the minimum column sum.
p = 1: the maximum column sum.
p = -1: the minimum column sum.
p = 2: the 2-norm, equal to the maximum singular value.
p = -2: the minimum singular value.
ALGORITHM:
Computation is performed by the cond() function of the SciPy/NumPy library.
EXAMPLES:
First over the reals.
sage: A = matrix(RDF, 4, [(1/4)*x^3 for x in range(16)]); A
[ 0.0 0.25 2.0 6.75]
[ 16.0 31.25 54.0 85.75]
[ 128.0 182.25 250.0 332.75]
[ 432.0 549.25 686.0 843.75]
sage: A.condition()
9923.88955...
sage: A.condition(p='frob')
9923.88955...
sage: A.condition(p=Infinity)
22738.5
sage: A.condition(p=-Infinity)
17.5
sage: A.condition(p=1)
12139.21...
sage: A.condition(p=-1)
550.0
sage: A.condition(p=2)
9897.8088...
sage: A.condition(p=-2)
0.000101032462...
And over the complex numbers.
sage: B = matrix(CDF, 3, [x + x^2*I for x in range(9)]); B
[ 0.0 1.0 + 1.0*I 2.0 + 4.0*I]
[ 3.0 + 9.0*I 4.0 + 16.0*I 5.0 + 25.0*I]
[6.0 + 36.0*I 7.0 + 49.0*I 8.0 + 64.0*I]
sage: B.condition()
203.851798...
sage: B.condition(p='frob')
203.851798...
sage: B.condition(p=Infinity)
369.55630...
sage: B.condition(p=-Infinity)
5.46112969...
sage: B.condition(p=1)
289.251481...
sage: B.condition(p=-1)
20.4566639...
sage: B.condition(p=2)
202.653543...
sage: B.condition(p=-2)
0.00493453005...
Hilbert matrices are famously ill-conditioned, while an identity matrix can hit the minimum with the right norm.
sage: A = matrix(RDF, 10, [1/(i+j+1) for i in range(10) for j in range(10)])
sage: A.condition()
1.633...e+13
sage: id = identity_matrix(CDF, 10)
sage: id.condition(p=1)
1.0
Return values are in \(RDF\).
sage: A = matrix(CDF, 2, range(1,5))
sage: A.condition() in RDF
True
Rectangular and singular matrices raise errors if p is not ‘sv’.
sage: A = matrix(RDF, 2, 3, range(6))
sage: A.condition()
Traceback (most recent call last):
...
TypeError: matrix must be square if p is not 'sv', not 2 x 3
sage: A.condition('sv')
7.34...
sage: A = matrix(QQ, 5, range(25))
sage: A.is_singular()
True
sage: B = A.change_ring(CDF)
sage: B.condition()
Traceback (most recent call last):
...
LinAlgError: Singular matrix
Improper values of p are caught.
sage: A = matrix(CDF, 2, range(1,5))
sage: A.condition(p='bogus')
Traceback (most recent call last):
...
ValueError: condition number 'p' must be +/- infinity, 'frob', 'sv' or an integer, not bogus
sage: A.condition(p=632)
Traceback (most recent call last):
...
ValueError: condition number integer values of 'p' must be -2, -1, 1 or 2, not 632
TESTS:
Some condition numbers, first by the definition which also exercises norm(), then by this method.
sage: A = matrix(CDF, [[1,2,4],[5,3,9],[7,8,6]])
sage: c = A.norm(2)*A.inverse().norm(2)
sage: d = A.condition(2)
sage: abs(c-d) < 1.0e-12
True
sage: c = A.norm(1)*A.inverse().norm(1)
sage: d = A.condition(1)
sage: abs(c-d) < 1.0e-12
True
Return the determinant of self.
ALGORITHM:
Use numpy
EXAMPLES:
sage: m = matrix(RDF,2,range(4)); m.det()
-2.0
sage: m = matrix(RDF,0,[]); m.det()
1.0
sage: m = matrix(RDF, 2, range(6)); m.det()
Traceback (most recent call last):
...
ValueError: self must be a square matrix
Computes the left eigenspaces of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
Warning
This method returns eigenspaces that are all of dimension one, since it is impossible to ascertain if the numerical results belong to the same eigenspace. So this is deprecated in favor of the eigenmatrix routines, such as sage.matrix.matrix2.Matrix.eigenmatrix_right().
INPUT:
OUTPUT:
Return a list of pairs (e, V) where e is a (complex) eigenvalue and V is the associated left eigenspace as a vector space.
No attempt is made to determine if an eigenvalue has multiplicity greater than one, so all the eigenspaces returned have dimension one.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by creating an eigenspace with a canonical basis.
EXAMPLES:
This first test simply raises the deprecation warning.
sage: A = identity_matrix(RDF, 2)
sage: es = A.eigenspaces_left()
doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are
deprecated as of Sage version 5.0,
please use "eigenmatrix_left" instead
See http://trac.sagemath.org/11603 for details.
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
sage: spectrum = m.eigenspaces_left()
sage: spectrum[0][0]
2.0
sage: (RDF^4).subspace(spectrum[0][1].basis())
Vector space of degree 4 and dimension 1 over Real Double Field
Basis matrix:
[1.0 1.0 1.0 1.0]
sage: e, V = spectrum[2]
sage: v = V.basis()[0]
sage: diff = (v*m).change_ring(CDF) - e*v
sage: abs(abs(diff)) < 1e-14
True
TESTS:
sage: m.eigenspaces_left(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
Computes the right eigenspaces of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
Warning
This method returns eigenspaces that are all of dimension one, since it is impossible to ascertain if the numerical results belong to the same eigenspace. So this is deprecated in favor of the eigenmatrix routines, such as sage.matrix.matrix2.Matrix.eigenmatrix_right().
INPUT:
OUTPUT:
Return a list of pairs (e, V) where e is a (complex) eigenvalue and V is the associated right eigenspace as a vector space.
No attempt is made to determine if an eigenvalue has multiplicity greater than one, so all the eigenspaces returned have dimension one.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by creating an eigenspace with a canonical basis.
EXAMPLES:
This first test simply raises the deprecation warning.
sage: A = identity_matrix(RDF, 2)
sage: es = A.eigenspaces_right()
doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are
deprecated as of Sage version 5.0,
please use "eigenmatrix_right" instead
See http://trac.sagemath.org/11603 for details.
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
sage: m
[ -9.0 -14.0 19.0 -74.0]
[ -1.0 2.0 4.0 -11.0]
[ -4.0 -12.0 6.0 -32.0]
[ 0.0 -2.0 -1.0 1.0]
sage: spectrum = m.eigenspaces_right()
sage: spectrum[0][0]
2.0
sage: (RDF^4).subspace(spectrum[0][1].basis())
Vector space of degree 4 and dimension 1 over Real Double Field
Basis matrix:
[ 1.0 -2.0 3.0 1.0]
sage: e, V = spectrum[2]
sage: v = V.basis()[0]
sage: diff = (m*v).change_ring(CDF) - e*v
sage: abs(abs(diff)) < 3e-14
True
TESTS:
sage: m.eigenspaces_right(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
Returns a list of eigenvalues.
INPUT:
Warning
When using the 'symmetric' or 'hermitian' algorithms, no check is made on the input matrix, and only the entries below, and on, the main diagonal are employed in the computation.
Methods such as is_symmetric() and is_hermitian() could be used to verify this beforehand.
OUTPUT:
Default output for a square matrix of size \(n\) is a list of \(n\) eigenvalues from the complex double field, CDF. If the 'symmetric' or 'hermitian' algorithms are chosen, the returned eigenvalues are from the real double field, RDF.
If a tolerance is specified, an attempt is made to group eigenvalues that are numerically similar. The return is then a list of pairs, where each pair is an eigenvalue followed by its multiplicity. The eigenvalue reported is the mean of the eigenvalues computed, and these eigenvalues are contained in an interval (or disk) whose radius is less than 5*tol for \(n < 10,000\) in the worst case.
More precisely, for an \(n\times n\) matrix, the diameter of the interval containing similar eigenvalues could be as large as sum of the reciprocals of the first \(n\) integers times tol.
Warning
Use caution when using the tol parameter to group eigenvalues. See the examples below to see how this can go wrong.
EXAMPLES:
sage: m = matrix(RDF, 2, 2, [1,2,3,4])
sage: ev = m.eigenvalues(); ev
[-0.372281323..., 5.37228132...]
sage: ev[0].parent()
Complex Double Field
sage: m = matrix(RDF, 2, 2, [0,1,-1,0])
sage: m.eigenvalues(algorithm='default')
[1.0*I, -1.0*I]
sage: m = matrix(CDF, 2, 2, [I,1,-I,0])
sage: m.eigenvalues()
[-0.624810533... + 1.30024259...*I, 0.624810533... - 0.30024259...*I]
The adjacency matrix of a graph will be symmetric, and the eigenvalues will be real.
sage: A = graphs.PetersenGraph().adjacency_matrix()
sage: A = A.change_ring(RDF)
sage: ev = A.eigenvalues(algorithm='symmetric'); ev
[-2.0, -2.0, -2.0, -2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 3.0]
sage: ev[0].parent()
Real Double Field
The matrix A is “random”, but the construction of B provides a positive-definite Hermitian matrix. Note that the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a positive-definite matrix will be positive.
sage: A = matrix([[ 4*I + 5, 8*I + 1, 7*I + 5, 3*I + 5],
... [ 7*I - 2, -4*I + 7, -2*I + 4, 8*I + 8],
... [-2*I + 1, 6*I + 6, 5*I + 5, -I - 4],
... [ 5*I + 1, 6*I + 2, I - 4, -I + 3]])
sage: B = (A*A.conjugate_transpose()).change_ring(CDF)
sage: ev = B.eigenvalues(algorithm='hermitian'); ev
[2.68144025..., 49.5167998..., 274.086188..., 390.71557...]
sage: ev[0].parent()
Real Double Field
A tolerance can be given to aid in grouping eigenvalues that are similar numerically. However, if the parameter is too small it might split too finely. Too large, and it can go wrong very badly. Use with care.
sage: G = graphs.PetersenGraph()
sage: G.spectrum()
[3, 1, 1, 1, 1, 1, -2, -2, -2, -2]
sage: A = G.adjacency_matrix().change_ring(RDF)
sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5)
[(-2.0, 4), (1.0, 5), (3.0, 1)]
sage: A.eigenvalues(algorithm='symmetric', tol=2.5)
[(-2.0, 4), (1.33333333333, 6)]
An (extreme) example of properly grouping similar eigenvalues.
sage: G = graphs.HigmanSimsGraph()
sage: A = G.adjacency_matrix().change_ring(RDF)
sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5)
[(-8.0, 22), (2.0, 77), (22.0, 1)]
TESTS:
Testing bad input.
sage: A = matrix(CDF, 2, range(4))
sage: A.eigenvalues(algorithm='junk')
Traceback (most recent call last):
...
ValueError: algorithm must be 'default', 'symmetric', or 'hermitian', not junk
sage: A = matrix(CDF, 2, 3, range(6))
sage: A.eigenvalues()
Traceback (most recent call last):
...
ValueError: matrix must be square, not 2 x 3
sage: A = matrix(CDF, 2, [1, 2, 3, 4*I])
sage: A.eigenvalues(algorithm='symmetric')
Traceback (most recent call last):
...
TypeError: cannot apply symmetric algorithm to matrix with complex entries
sage: A = matrix(CDF, 2, 2, range(4))
sage: A.eigenvalues(tol='junk')
Traceback (most recent call last):
...
TypeError: tolerance parameter must be a real number, not junk
sage: A = matrix(CDF, 2, 2, range(4))
sage: A.eigenvalues(tol=-0.01)
Traceback (most recent call last):
...
ValueError: tolerance parameter must be positive, not -0.01
A very small matrix.
sage: matrix(CDF,0,0).eigenvalues()
[]
Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
OUTPUT: Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an associated left eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.
The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
EXAMPLES:
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
sage: m
[ -5.0 3.0 2.0 8.0]
[ 10.0 2.0 4.0 -2.0]
[ -1.0 -10.0 -10.0 -17.0]
[ -2.0 7.0 6.0 13.0]
sage: spectrum = m.left_eigenvectors()
sage: for i in range(len(spectrum)):
... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
sage: spectrum[0]
(2.0, [(1.0, 1.0, 1.0, 1.0)], 1)
sage: spectrum[1]
(1.0, [(1.0, 0.8, 0.8, 0.6)], 1)
sage: spectrum[2]
(-2.0, [(1.0, 0.4, 0.6, 0.2)], 1)
sage: spectrum[3]
(-1.0, [(1.0, 1.0, 2.0, 2.0)], 1)
Compute the right eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
OUTPUT:
Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an associated right eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.
The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
EXAMPLES:
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
sage: m
[ -9.0 -14.0 19.0 -74.0]
[ -1.0 2.0 4.0 -11.0]
[ -4.0 -12.0 6.0 -32.0]
[ 0.0 -2.0 -1.0 1.0]
sage: spectrum = m.right_eigenvectors()
sage: for i in range(len(spectrum)):
... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
sage: spectrum[0]
(2.0, [(1.0, -2.0, 3.0, 1.0)], 1)
sage: spectrum[1]
(1.0, [(1.0, -0.666666666667, 1.33333333333, 0.333333333333)], 1)
sage: spectrum[2]
(-2.0, [(1.0, -0.2, 1.0, 0.2)], 1)
sage: spectrum[3]
(-1.0, [(1.0, -0.5, 2.0, 0.5)], 1)
Calculate the exponential of this matrix X, which is the matrix
INPUT:
EXAMPLES:
sage: A=matrix(RDF, 2, [1,2,3,4]); A
[1.0 2.0]
[3.0 4.0]
sage: A.exp()
[51.9689561987 74.736564567]
[112.104846851 164.073803049]
sage: A.exp(algorithm='eig')
[51.9689561987 74.736564567]
[112.104846851 164.073803049]
sage: A.exp(algorithm='taylor', order=5)
[19.9583333333 28.0833333333]
[ 42.125 62.0833333333]
sage: A.exp(algorithm='taylor')
[51.9689035511 74.7364878369]
[112.104731755 164.073635306]
sage: A=matrix(CDF, 2, [1,2+I,3*I,4]); A
[ 1.0 2.0 + 1.0*I]
[ 3.0*I 4.0]
sage: A.exp()
[-19.6146029538 + 12.5177438468*I 3.79496364496 + 28.8837993066*I]
[-32.3835809809 + 21.8842359579*I 2.26963300409 + 44.9013248277*I]
sage: A.exp(algorithm='eig')
[-19.6146029538 + 12.5177438468*I 3.79496364496 + 28.8837993066*I]
[-32.3835809809 + 21.8842359579*I 2.26963300409 + 44.9013248277*I]
sage: A.exp(algorithm='taylor', order=5)
[ -6.29166666667 + 14.25*I 14.0833333333 + 15.7916666667*I]
[ -10.5 + 26.375*I 20.0833333333 + 24.75*I]
sage: A.exp(algorithm='taylor')
[-19.6146006163 + 12.5177432169*I 3.79496442472 + 28.8837964828*I]
[-32.3835771246 + 21.8842351994*I 2.26963458304 + 44.9013203415*I]
Returns True if the matrix is equal to its conjugate-transpose.
INPUT:
OUTPUT:
True if the matrix is square and equal to the transpose with every entry conjugated, and False otherwise.
Note that if conjugation has no effect on elements of the base ring (such as for integers), then the is_symmetric() method is equivalent and faster.
The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares corresponing entries on either side of the diagonal (and on the diagonal itself) to see if they are conjugates, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the schur() method) and checks that the result is a diagonal matrix with real entries.
So the naive algorithm can finish quickly for a matrix that is not Hermitian, while the orthonormal algorithm will always compute a Schur decomposition before going through a similar check of the matrix entry-by-entry.
EXAMPLES:
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: A.is_hermitian(algorithm='orthonormal')
False
sage: A.is_hermitian(algorithm='naive')
False
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian(algorithm='orthonormal')
True
sage: B.is_hermitian(algorithm='naive')
True
A matrix that is nearly Hermitian, but for one non-real diagonal entry.
sage: A = matrix(CDF, [[ 2, 2-I, 1+4*I],
... [ 2+I, 3+I, 2-6*I],
... [1-4*I, 2+6*I, 5]])
sage: A.is_hermitian(algorithm='orthonormal')
False
sage: A[1,1] = 132
sage: A.is_hermitian(algorithm='orthonormal')
True
We get a unitary matrix from the SVD routine and use this numerical matrix to create a matrix that should be Hermitian (indeed it should be the identity matrix), but with some imprecision. We use this to illustrate that if the tolerance is set too small, then we can be too strict about the equality of entries and may achieve the wrong result (depending on the system):
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: U, _, _ = A.SVD()
sage: B=U*U.conjugate_transpose()
sage: B.is_hermitian(algorithm='naive')
True
sage: B.is_hermitian(algorithm='naive', tol=1.0e-17) # random
False
sage: B.is_hermitian(algorithm='naive', tol=1.0e-15)
True
A square, empty matrix is trivially Hermitian.
sage: A = matrix(RDF, 0, 0)
sage: A.is_hermitian()
True
Rectangular matrices are never Hermitian, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4)
sage: A.is_hermitian()
False
TESTS:
The tolerance must be strictly positive.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_hermitian(tol = -3.1)
Traceback (most recent call last):
...
ValueError: tolerance must be positive, not -3.1
The algorithm keyword gets checked.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_hermitian(algorithm='junk')
Traceback (most recent call last):
...
ValueError: algorithm must be 'naive' or 'orthonormal', not junk
AUTHOR:
Returns True if the matrix commutes with its conjugate-transpose.
INPUT:
OUTPUT:
True if the matrix is square and commutes with its conjugate-transpose, and False otherwise.
Normal matrices are precisely those that can be diagonalized by a unitary matrix.
The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares entries of the two possible products of the matrix with its conjugate-transpose, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the schur() method) and checks that the result is a diagonal matrix. An orthonormal diagonalization is equivalent to being normal.
So the naive algorithm can finish fairly quickly for a matrix that is not normal, once the products have been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.
EXAMPLES:
First over the complexes. B is Hermitian, hence normal.
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian()
True
sage: B.is_normal(algorithm='orthonormal')
True
sage: B.is_normal(algorithm='naive')
True
sage: B[0,0] = I
sage: B.is_normal(algorithm='orthonormal')
False
sage: B.is_normal(algorithm='naive')
False
Now over the reals. Circulant matrices are normal.
sage: G = graphs.CirculantGraph(20, [3, 7])
sage: D = digraphs.Circuit(20)
sage: A = 3*D.adjacency_matrix() - 5*G.adjacency_matrix()
sage: A = A.change_ring(RDF)
sage: A.is_normal()
True
sage: A.is_normal(algorithm = 'naive')
True
sage: A[19,0] = 4.0
sage: A.is_normal()
False
sage: A.is_normal(algorithm = 'naive')
False
Skew-Hermitian matrices are normal.
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: B = A - A.conjugate_transpose()
sage: B.is_hermitian()
False
sage: B.is_normal()
True
sage: B.is_normal(algorithm='naive')
True
A small matrix that does not fit into any of the usual categories of normal matrices.
sage: A = matrix(RDF, [[1, -1],
... [1, 1]])
sage: A.is_normal()
True
sage: not A.is_hermitian() and not A.is_skew_symmetric()
True
Sage has several fields besides the entire complex numbers where conjugation is non-trivial.
sage: F.<b> = QuadraticField(-7)
sage: C = matrix(F, [[-2*b - 3, 7*b - 6, -b + 3],
... [-2*b - 3, -3*b + 2, -2*b],
... [ b + 1, 0, -2]])
sage: C = C*C.conjugate_transpose()
sage: C.is_normal()
True
A square, empty matrix is trivially normal.
sage: A = matrix(CDF, 0, 0)
sage: A.is_normal()
True
Rectangular matrices are never normal, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4)
sage: A.is_normal()
False
TESTS:
The tolerance must be strictly positive.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_normal(tol = -3.1)
Traceback (most recent call last):
...
ValueError: tolerance must be positive, not -3.1
The algorithm keyword gets checked.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_normal(algorithm='junk')
Traceback (most recent call last):
...
ValueError: algorithm must be 'naive' or 'orthonormal', not junk
AUTHOR:
- Rob Beezer (2011-03-31)
Determines if a matrix is positive definite.
A matrix \(A\) is positive definite if it is square, is Hermitian (which reduces to symmetric in the real case), and for every nonzero vector \(\vec{x}\),
where \(\vec{x}^\ast\) is the conjugate-transpose in the complex case and just the transpose in the real case. Equivalently, a positive definite matrix has only positive eigenvalues and only positive determinants of leading principal submatrices.
INPUT:
Any matrix over RDF or CDF.
OUTPUT:
True if and only if the matrix is square, Hermitian, and meets the condition above on the quadratic form. The result is cached.
IMPLEMENTATION:
The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the cholesky() method compute and cache both the Cholesky decomposition and the positive-definiteness. So the is_positive_definite() method or catching a ValueError from the cholesky() method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.
EXAMPLES:
A matrix over RDF that is positive definite.
sage: M = matrix(RDF,[[ 1, 1, 1, 1, 1],
... [ 1, 5, 31, 121, 341],
... [ 1, 31, 341, 1555, 4681],
... [ 1,121, 1555, 7381, 22621],
... [ 1,341, 4681, 22621, 69905]])
sage: M.is_symmetric()
True
sage: M.eigenvalues()
[77547.66..., 82.44..., 2.41..., 0.46..., 0.011...]
sage: [round(M[:i,:i].determinant()) for i in range(1, M.nrows()+1)]
[1, 4, 460, 27936, 82944]
sage: M.is_positive_definite()
True
A matrix over CDF that is positive definite.
sage: C = matrix(CDF, [[ 23, 17*I + 3, 24*I + 25, 21*I],
... [ -17*I + 3, 38, -69*I + 89, 7*I + 15],
... [-24*I + 25, 69*I + 89, 976, 24*I + 6],
... [ -21*I, -7*I + 15, -24*I + 6, 28]])
sage: C.is_hermitian()
True
sage: [x.real() for x in C.eigenvalues()]
[991.46..., 55.96..., 3.69..., 13.87...]
sage: [round(C[:i,:i].determinant().real()) for i in range(1, C.nrows()+1)]
[23, 576, 359540, 2842600]
sage: C.is_positive_definite()
True
A matrix over RDF that is not positive definite.
sage: A = matrix(RDF, [[ 3, -6, 9, 6, -9],
... [-6, 11, -16, -11, 17],
... [ 9, -16, 28, 16, -40],
... [ 6, -11, 16, 9, -19],
... [-9, 17, -40, -19, 68]])
sage: A.is_symmetric()
True
sage: A.eigenvalues()
[108.07..., 13.02..., -0.02..., -0.70..., -1.37...]
sage: [round(A[:i,:i].determinant()) for i in range(1, A.nrows()+1)]
[3, -3, -15, 30, -30]
sage: A.is_positive_definite()
False
A matrix over CDF that is not positive definite.
sage: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I],
... [4 + 2*I, 8, 10*I],
... [2 - 2*I, -10*I, -3]])
sage: B.is_hermitian()
True
sage: [ev.real() for ev in B.eigenvalues()]
[15.88..., 0.08..., -8.97...]
sage: [round(B[:i,:i].determinant().real()) for i in range(1, B.nrows()+1)]
[2, -4, -12]
sage: B.is_positive_definite()
False
A large random matrix that is guaranteed by theory to be positive definite.
sage: R = random_matrix(CDF, 200)
sage: H = R.conjugate_transpose()*R
sage: H.is_positive_definite()
True
TESTS:
A trivially small case.
sage: S = matrix(CDF, [])
sage: S.nrows(), S.ncols()
(0, 0)
sage: S.is_positive_definite()
True
A rectangular matrix will never be positive definite.
sage: R = matrix(RDF, 2, 3, range(6))
sage: R.is_positive_definite()
False
A non-Hermitian matrix will never be positive definite.
sage: T = matrix(CDF, 8, 8, range(64))
sage: T.is_positive_definite()
False
AUTHOR:
Return whether this matrix is symmetric, to the given tolerance.
EXAMPLES:
sage: m = matrix(RDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: m.is_symmetric()
False
sage: m[1,0] = 1.1; m
[0.0 1.0]
[1.1 3.0]
sage: m.is_symmetric()
False
Returns True if the columns of the matrix are an orthonormal basis.
For a matrix with real entries this determines if a matrix is “orthogonal” and for a matrix with complex entries this determines if the matrix is “unitary.”
INPUT:
OUTPUT:
True if the matrix is square and its conjugate-transpose is its inverse, and False otherwise. In other words, a matrix is orthogonal or unitary if the product of its conjugate-transpose times the matrix is the identity matrix.
The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply computes the product of the conjugate-transpose with the matrix and compares the entries to the identity matrix, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the schur() method) and checks that the result is a diagonal matrix with entries of modulus 1, which is equivalent to being unitary.
So the naive algorithm might finish fairly quickly for a matrix that is not unitary, once the product has been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.
EXAMPLES:
A matrix that is far from unitary.
sage: A = matrix(RDF, 4, range(16))
sage: A.conjugate().transpose()*A
[224.0 248.0 272.0 296.0]
[248.0 276.0 304.0 332.0]
[272.0 304.0 336.0 368.0]
[296.0 332.0 368.0 404.0]
sage: A.is_unitary()
False
sage: A.is_unitary(algorithm='naive')
False
sage: A.is_unitary(algorithm='orthonormal')
False
The QR decoposition will produce a unitary matrix as Q and the SVD decomposition will create two unitary matrices, U and V.
sage: A = matrix(CDF, [[ 1 - I, -3*I, -2 + I, 1, -2 + 3*I],
... [ 1 - I, -2 + I, 1 + 4*I, 0, 2 + I],
... [ -1, -5 + I, -2 + I, 1 + I, -5 - 4*I],
... [-2 + 4*I, 2 - I, 8 - 4*I, 1 - 8*I, 3 - 2*I]])
sage: Q, R = A.QR()
sage: Q.is_unitary()
True
sage: U, S, V = A.SVD()
sage: U.is_unitary(algorithm='naive')
True
sage: U.is_unitary(algorithm='orthonormal')
True
sage: V.is_unitary(algorithm='naive') # not tested - known bug (trac #11248)
True
If we make the tolerance too strict we can get misleading results.
sage: A = matrix(RDF, 10, 10, [1/(i+j+1) for i in range(10) for j in range(10)])
sage: Q, R = A.QR()
sage: Q.is_unitary(algorithm='naive', tol=1e-16)
False
sage: Q.is_unitary(algorithm='orthonormal', tol=1e-17)
False
Rectangular matrices are not unitary/orthogonal, even if their columns form an orthonormal set.
sage: A = matrix(CDF, [[1,0], [0,0], [0,1]])
sage: A.is_unitary()
False
The smallest cases. The Schur decomposition used by the orthonormal algorithm will fail on a matrix of size zero.
sage: P = matrix(CDF, 0, 0)
sage: P.is_unitary(algorithm='naive')
True
sage: P = matrix(CDF, 1, 1, [1])
sage: P.is_unitary(algorithm='orthonormal')
True
sage: P = matrix(CDF, 0, 0,)
sage: P.is_unitary(algorithm='orthonormal')
Traceback (most recent call last):
...
ValueError: failed to create intent(cache|hide)|optional array-- must have defined dimensions but got (0,)
TESTS:
sage: P = matrix(CDF, 2, 2)
sage: P.is_unitary(tol='junk')
Traceback (most recent call last):
...
TypeError: tolerance must be a real number, not junk
sage: P.is_unitary(tol=-0.3)
Traceback (most recent call last):
...
ValueError: tolerance must be positive, not -0.3
sage: P.is_unitary(algorithm='junk')
Traceback (most recent call last):
...
ValueError: algorithm must be 'naive' or 'orthonormal', not junk
AUTHOR:
Computes the left eigenspaces of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
Warning
This method returns eigenspaces that are all of dimension one, since it is impossible to ascertain if the numerical results belong to the same eigenspace. So this is deprecated in favor of the eigenmatrix routines, such as sage.matrix.matrix2.Matrix.eigenmatrix_right().
INPUT:
OUTPUT:
Return a list of pairs (e, V) where e is a (complex) eigenvalue and V is the associated left eigenspace as a vector space.
No attempt is made to determine if an eigenvalue has multiplicity greater than one, so all the eigenspaces returned have dimension one.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by creating an eigenspace with a canonical basis.
EXAMPLES:
This first test simply raises the deprecation warning.
sage: A = identity_matrix(RDF, 2)
sage: es = A.eigenspaces_left()
doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are
deprecated as of Sage version 5.0,
please use "eigenmatrix_left" instead
See http://trac.sagemath.org/11603 for details.
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
sage: spectrum = m.eigenspaces_left()
sage: spectrum[0][0]
2.0
sage: (RDF^4).subspace(spectrum[0][1].basis())
Vector space of degree 4 and dimension 1 over Real Double Field
Basis matrix:
[1.0 1.0 1.0 1.0]
sage: e, V = spectrum[2]
sage: v = V.basis()[0]
sage: diff = (v*m).change_ring(CDF) - e*v
sage: abs(abs(diff)) < 1e-14
True
TESTS:
sage: m.eigenspaces_left(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
OUTPUT: Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an associated left eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.
The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
EXAMPLES:
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
sage: m
[ -5.0 3.0 2.0 8.0]
[ 10.0 2.0 4.0 -2.0]
[ -1.0 -10.0 -10.0 -17.0]
[ -2.0 7.0 6.0 13.0]
sage: spectrum = m.left_eigenvectors()
sage: for i in range(len(spectrum)):
... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
sage: spectrum[0]
(2.0, [(1.0, 1.0, 1.0, 1.0)], 1)
sage: spectrum[1]
(1.0, [(1.0, 0.8, 0.8, 0.6)], 1)
sage: spectrum[2]
(-2.0, [(1.0, 0.4, 0.6, 0.2)], 1)
sage: spectrum[3]
(-1.0, [(1.0, 1.0, 2.0, 2.0)], 1)
Compute the log of the absolute value of the determinant using LU decomposition.
Note
This is useful if the usual determinant overflows.
EXAMPLES:
sage: m = matrix(RDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: RDF(log(abs(m.determinant())))
0.69314718056
sage: m.log_determinant()
0.69314718056
sage: m = matrix(RDF,0,0,[]); m
[]
sage: m.log_determinant()
0.0
sage: m = matrix(CDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: RDF(log(abs(m.determinant())))
0.69314718056
sage: m.log_determinant()
0.69314718056
sage: m = matrix(CDF,0,0,[]); m
[]
sage: m.log_determinant()
0.0
Returns the norm of the matrix.
INPUT:
OUTPUT:
Returned value is a double precision floating point value in RDF. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.
p = 'frob': the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes
p = Infinity or p = oo: the maximum row sum.
p = -Infinity or p = -oo: the minimum column sum.
p = 1: the maximum column sum.
p = -1: the minimum column sum.
p = 2: the induced 2-norm, equal to the maximum singular value.
p = -2: the minimum singular value.
ALGORITHM:
Computation is performed by the norm() function of the SciPy/NumPy library.
EXAMPLES:
First over the reals.
sage: A = matrix(RDF, 3, range(-3, 6)); A
[-3.0 -2.0 -1.0]
[ 0.0 1.0 2.0]
[ 3.0 4.0 5.0]
sage: A.norm()
doctest:...: DeprecationWarning: The default norm will be changing from p='frob' to p=2. Use p='frob' explicitly to continue calculating the Frobenius norm.
See http://trac.sagemath.org/13643 for details.
8.30662386...
sage: A.norm(p='frob')
8.30662386...
sage: A.norm(p=Infinity)
12.0
sage: A.norm(p=-Infinity)
3.0
sage: A.norm(p=1)
8.0
sage: A.norm(p=-1)
6.0
sage: A.norm(p=2)
7.99575670...
sage: A.norm(p=-2) < 10^-15
True
And over the complex numbers.
sage: B = matrix(CDF, 2, [[1+I, 2+3*I],[3+4*I,3*I]]); B
[1.0 + 1.0*I 2.0 + 3.0*I]
[3.0 + 4.0*I 3.0*I]
sage: B.norm()
7.0
sage: B.norm(p='frob')
7.0
sage: B.norm(p=Infinity)
8.0
sage: B.norm(p=-Infinity)
5.01976483...
sage: B.norm(p=1)
6.60555127...
sage: B.norm(p=-1)
6.41421356...
sage: B.norm(p=2)
6.66189877...
sage: B.norm(p=-2)
2.14921023...
Since it is invariant under unitary multiplication, the Frobenius norm is equal to the square root of the sum of squares of the singular values.
sage: A = matrix(RDF, 5, range(1,26))
sage: f = A.norm(p='frob')
sage: U, S, V = A.SVD()
sage: s = sqrt(sum([S[i,i]^2 for i in range(5)]))
sage: abs(f-s) < 1.0e-12
True
Return values are in \(RDF\).
sage: A = matrix(CDF, 2, range(4))
sage: A.norm() in RDF
True
Improper values of p are caught.
sage: A.norm(p='bogus')
Traceback (most recent call last):
...
ValueError: matrix norm 'p' must be +/- infinity, 'frob' or an integer, not bogus
sage: A.norm(p=632)
Traceback (most recent call last):
...
ValueError: matrix norm integer values of 'p' must be -2, -1, 1 or 2, not 632
This method returns a copy of the matrix as a numpy array. It uses the numpy C/api so is very fast.
INPUT:
EXAMPLES:
sage: m = matrix(RDF,[[1,2],[3,4]])
sage: n = m.numpy()
sage: import numpy
sage: numpy.linalg.eig(n)
(array([-0.37228132, 5.37228132]), array([[-0.82456484, -0.41597356],
[ 0.56576746, -0.90937671]]))
sage: m = matrix(RDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: m.numpy()
array([[ 0., 1., 2.],
[ 3., 4., 5.]])
Alternatively, numpy automatically calls this function (via the magic __array__() method) to convert Sage matrices to numpy arrays:
sage: import numpy
sage: m = matrix(RDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: numpy.array(m)
array([[ 0., 1., 2.],
[ 3., 4., 5.]])
sage: numpy.array(m).dtype
dtype('float64')
sage: m = matrix(CDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: numpy.array(m)
array([[ 0.+0.j, 1.+0.j, 2.+0.j],
[ 3.+0.j, 4.+0.j, 5.+0.j]])
sage: numpy.array(m).dtype
dtype('complex128')
TESTS:
sage: m = matrix(RDF,0,5,[]); m
[]
sage: m.numpy()
array([], shape=(0, 5), dtype=float64)
sage: m = matrix(RDF,5,0,[]); m
[]
sage: m.numpy()
array([], shape=(5, 0), dtype=float64)
Computes the right eigenspaces of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
Warning
This method returns eigenspaces that are all of dimension one, since it is impossible to ascertain if the numerical results belong to the same eigenspace. So this is deprecated in favor of the eigenmatrix routines, such as sage.matrix.matrix2.Matrix.eigenmatrix_right().
INPUT:
OUTPUT:
Return a list of pairs (e, V) where e is a (complex) eigenvalue and V is the associated right eigenspace as a vector space.
No attempt is made to determine if an eigenvalue has multiplicity greater than one, so all the eigenspaces returned have dimension one.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by creating an eigenspace with a canonical basis.
EXAMPLES:
This first test simply raises the deprecation warning.
sage: A = identity_matrix(RDF, 2)
sage: es = A.eigenspaces_right()
doctest:...: DeprecationWarning: Eigenspaces of RDF/CDF matrices are
deprecated as of Sage version 5.0,
please use "eigenmatrix_right" instead
See http://trac.sagemath.org/11603 for details.
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
sage: m
[ -9.0 -14.0 19.0 -74.0]
[ -1.0 2.0 4.0 -11.0]
[ -4.0 -12.0 6.0 -32.0]
[ 0.0 -2.0 -1.0 1.0]
sage: spectrum = m.eigenspaces_right()
sage: spectrum[0][0]
2.0
sage: (RDF^4).subspace(spectrum[0][1].basis())
Vector space of degree 4 and dimension 1 over Real Double Field
Basis matrix:
[ 1.0 -2.0 3.0 1.0]
sage: e, V = spectrum[2]
sage: v = V.basis()[0]
sage: diff = (m*v).change_ring(CDF) - e*v
sage: abs(abs(diff)) < 3e-14
True
TESTS:
sage: m.eigenspaces_right(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
Compute the right eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
OUTPUT:
Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an associated right eigenvector. If the matrix is of size \(n\), then there are \(n\) triples. Values are computed with the SciPy library.
The format of this output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
EXAMPLES:
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]])
sage: m
[ -9.0 -14.0 19.0 -74.0]
[ -1.0 2.0 4.0 -11.0]
[ -4.0 -12.0 6.0 -32.0]
[ 0.0 -2.0 -1.0 1.0]
sage: spectrum = m.right_eigenvectors()
sage: for i in range(len(spectrum)):
... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
sage: spectrum[0]
(2.0, [(1.0, -2.0, 3.0, 1.0)], 1)
sage: spectrum[1]
(1.0, [(1.0, -0.666666666667, 1.33333333333, 0.333333333333)], 1)
sage: spectrum[2]
(-2.0, [(1.0, -0.2, 1.0, 0.2)], 1)
sage: spectrum[3]
(-1.0, [(1.0, -0.5, 2.0, 0.5)], 1)
Returns a copy of the matrix where all entries have been rounded to a given precision in decimal digits (default 0 digits).
INPUT:
OUTPUT:
A modified copy of the matrix
EXAMPLES:
sage: M=matrix(CDF, [[10.234r + 34.2343jr, 34e10r]])
sage: M
[10.234 + 34.2343*I 3.4e+11]
sage: M.round(2)
[10.23 + 34.23*I 3.4e+11]
sage: M.round()
[10.0 + 34.0*I 3.4e+11]
Returns the Schur decomposition of the matrix.
INPUT:
OUTPUT:
A pair of immutable matrices. The first is a unitary matrix \(Q\). The second, \(T\), is upper-triangular when returned over the complex numbers, while it is almost upper-triangular over the reals. In the latter case, there can be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues of self.
If self is the matrix \(A\), then
where the latter matrix is the conjugate-transpose of Q, which is also the inverse of Q, since Q is unitary.
Note that in the case of a normal matrix (Hermitian, symmetric, and others), the upper-triangular matrix is a diagonal matrix with eigenvalues of self on the diagonal, and the unitary matrix has columns that form an orthonormal basis composed of eigenvectors of self. This is known as “orthonormal diagonalization”.
Warning
The Schur decomposition is not unique, as there may be numerous choices for the vectors of the orthonormal basis, and consequently different possibilities for the upper-triangular matrix. However, the diagonal of the upper-triangular matrix will always contain the eigenvalues of the matrix (in the complex version), or \(2\times 2\) block matrices in the real version representing pairs of conjugate complex eigenvalues.
In particular, results may vary across systems and processors.
EXAMPLES:
First over the complexes. The similar matrix is always upper-triangular in this case.
sage: A = matrix(CDF, 4, 4, range(16)) + matrix(CDF, 4, 4, [x^3*I for x in range(0, 16)])
sage: Q, T = A.schur()
sage: (Q*Q.conjugate().transpose()).zero_at(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
True
sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
sage: eigenvalues = [T[i,i] for i in range(4)]; eigenvalues
[30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I]
sage: A.eigenvalues()
[30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I]
sage: abs(A.norm()-T.norm()) < 1e-10
True
We begin with a real matrix but ask for a decomposition over the complexes. The result will yield an upper-triangular matrix over the complex numbers for T.
sage: A = matrix(RDF, 4, 4, [x^3 for x in range(16)])
sage: Q, T = A.schur(base_ring=CDF)
sage: (Q*Q.conjugate().transpose()).zero_at(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: T.parent()
Full MatrixSpace of 4 by 4 dense matrices over Complex Double Field
sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
True
sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
Now totally over the reals. But with complex eigenvalues, the similar matrix may not be upper-triangular. But “at worst” there may be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues. These blocks will then just interrupt the zeros below the main diagonal. This example has a pair of these of the blocks.
sage: A = matrix(RDF, 4, 4, [[1, 0, -3, -1],
... [4, -16, -7, 0],
... [1, 21, 1, -2],
... [26, -1, -2, 1]])
sage: Q, T = A.schur()
sage: (Q*Q.conjugate().transpose()).zero_at(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)])
False
sage: all([T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i-1)])
True
sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
sage: sorted(T[0:2,0:2].eigenvalues() + T[2:4,2:4].eigenvalues())
[-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I]
sage: sorted(A.eigenvalues())
[-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I]
sage: abs(A.norm()-T.norm()) < 1e-12
True
Starting with complex numbers and requesting a result over the reals will never happen.
sage: A = matrix(CDF, 2, 2, [[2+I, -1+3*I], [5-4*I, 2-7*I]])
sage: A.schur(base_ring=RDF)
Traceback (most recent call last):
...
TypeError: unable to convert input matrix over CDF to a matrix over RDF
If theory predicts your matrix is real, but it contains some very small imaginary parts, you can specify the cutoff for “small” imaginary parts, then request the output as real matrices, and let the routine do the rest.
sage: A = matrix(RDF, 2, 2, [1, 1, -1, 0]) + matrix(CDF, 2, 2, [1.0e-14*I]*4)
sage: B = A.zero_at(1.0e-12)
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field
sage: Q, T = B.schur(RDF)
sage: Q.parent()
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
sage: T.parent()
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
sage: Q.round(6)
[ 0.707107 0.707107]
[-0.707107 0.707107]
sage: T.round(6)
[ 0.5 1.5]
[-0.5 0.5]
sage: (Q*T*Q.conjugate().transpose()-B).zero_at(1.0e-11)
[0.0 0.0]
[0.0 0.0]
A Hermitian matrix has real eigenvalues, so the similar matrix will be upper-triangular. Furthermore, a Hermitian matrix is diagonalizable with respect to an orthonormal basis, composed of eigenvectors of the matrix. Here that basis is the set of columns of the unitary matrix.
sage: A = matrix(CDF, [[ 52, -9*I - 8, 6*I - 187, -188*I + 2],
... [ 9*I - 8, 12, -58*I + 59, 30*I + 42],
... [-6*I - 187, 58*I + 59, 2677, 2264*I + 65],
... [ 188*I + 2, -30*I + 42, -2264*I + 65, 2080]])
sage: Q, T = A.schur()
sage: T = T.zero_at(1.0e-12).change_ring(RDF)
sage: T.round(6)
[4680.13301 0.0 0.0 0.0]
[ 0.0 102.715967 0.0 0.0]
[ 0.0 0.0 35.039344 0.0]
[ 0.0 0.0 0.0 3.11168]
sage: (Q*Q.conjugate().transpose()).zero_at(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: (Q*T*Q.conjugate().transpose()-A).zero_at(1.0e-11)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
Similarly, a real symmetric matrix has only real eigenvalues, and there is an orthonormal basis composed of eigenvectors of the matrix.
sage: A = matrix(RDF, [[ 1, -2, 5, -3],
... [-2, 9, 1, 5],
... [ 5, 1, 3 , 7],
... [-3, 5, 7, -8]])
sage: Q, T = A.schur()
sage: Q.round(4)
[-0.3027 -0.751 0.576 -0.1121]
[ 0.139 -0.3892 -0.2648 0.8713]
[ 0.4361 0.359 0.7599 0.3217]
[ -0.836 0.3945 0.1438 0.3533]
sage: T = T.zero_at(10^-12)
sage: all(abs(e) < 10^-4 for e in (T - diagonal_matrix(RDF, [-13.5698, -0.8508, 7.7664, 11.6542])).list())
True
sage: (Q*Q.transpose()).zero_at(1.0e-12)
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 1.0 0.0]
[0.0 0.0 0.0 1.0]
sage: (Q*T*Q.transpose()-A).zero_at(1.0e-11)
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
[0.0 0.0 0.0 0.0]
The results are cached, both as a real factorization and also as a complex factorization. This means the returned matrices are immutable.
sage: A = matrix(RDF, 2, 2, [[0, -1], [1, 0]])
sage: Qr, Tr = A.schur(base_ring=RDF)
sage: Qc, Tc = A.schur(base_ring=CDF)
sage: all([M.is_immutable() for M in [Qr, Tr, Qc, Tc]])
True
sage: Tr.round(6) != Tc.round(6)
True
TESTS:
The Schur factorization is only defined for square matrices.
sage: A = matrix(RDF, 4, 5, range(20))
sage: A.schur()
Traceback (most recent call last):
...
ValueError: Schur decomposition requires a square matrix, not a 4 x 5 matrix
A base ring request is checked.
sage: A = matrix(RDF, 3, range(9))
sage: A.schur(base_ring=QQ)
Traceback (most recent call last):
...
ValueError: base ring of Schur decomposition matrices must be RDF or CDF, not Rational Field
AUTHOR:
Returns a sorted list of the singular values of the matrix.
INPUT:
OUTPUT:
A sorted list of the singular values of the matrix, which are the diagonal entries of the “S” matrix in the SVD decomposition. As such, the values are real and are returned as elements of RDF. The list is sorted with larger values first, and since theory predicts these values are always positive, for a rank-deficient matrix the list should end in zeros (but in practice may not). The length of the list is the minimum of the row count and column count for the matrix.
The number of non-zero singular values will be the rank of the matrix. However, as a numerical matrix, it is impossible to control the difference between zero entries and very small non-zero entries. As an informed consumer it is up to you to use the output responsibly. We will do our best, and give you the tools to work with the output, but we cannot give you a guarantee.
With eps set to None you will get the raw singular values and can manage them as you see fit. You may also set eps to any positive floating point value you wish. If you set eps to ‘auto’ this routine will compute a reasonable cutoff value, based on the size of the matrix, the largest singular value and the smallest nonzero value representable by the 53-bit precision values used. See the discussion at page 268 of [WATKINS].
See the examples for a way to use the “verbose” facility to easily watch the zero cutoffs in action.
ALGORITHM:
The singular values come from the SVD decomposition computed by SciPy/NumPy.
EXAMPLES:
Singular values close to zero have trailing digits that may vary on different hardware. For exact matrices, the number of non-zero singular values will equal the rank of the matrix. So for some of the doctests we round the small singular values that ideally would be zero, to control the variability across hardware.
This matrix has a determinant of one. A chain of two or three theorems implies the product of the singular values must also be one.
sage: A = matrix(QQ, [[ 1, 0, 0, 0, 0, 1, 3],
... [-2, 1, 1, -2, 0, -4, 0],
... [ 1, 0, 1, -4, -6, -3, 7],
... [-2, 2, 1, 1, 7, 1, -1],
... [-1, 0, -1, 5, 8, 4, -6],
... [ 4, -2, -2, 1, -3, 0, 8],
... [-2, 1, 0, 2, 7, 3, -4]])
sage: A.determinant()
1
sage: B = A.change_ring(RDF)
sage: sv = B.singular_values(); sv
[20.5239806589, 8.48683702854, 5.86168134845, 2.44291658993,
0.583197014472, 0.269332872866, 0.00255244880761]
sage: prod(sv)
1.0
An exact matrix that is obviously not of full rank, and then a computation of the singular values after conversion to an approximate matrix.
sage: A = matrix(QQ, [[1/3, 2/3, 11/3],
... [2/3, 1/3, 7/3],
... [2/3, 5/3, 27/3]])
sage: A.rank()
2
sage: B = A.change_ring(CDF)
sage: sv = B.singular_values()
sage: sv[0:2]
[10.1973039..., 0.487045871...]
sage: sv[2] < 1e-14
True
A matrix of rank 3 over the complex numbers.
sage: A = matrix(CDF, [[46*I - 28, -47*I - 50, 21*I + 51, -62*I - 782, 13*I + 22],
... [35*I - 20, -32*I - 46, 18*I + 43, -57*I - 670, 7*I + 3],
... [22*I - 13, -23*I - 23, 9*I + 24, -26*I - 347, 7*I + 13],
... [-44*I + 23, 41*I + 57, -19*I - 54, 60*I + 757, -11*I - 9],
... [30*I - 18, -30*I - 34, 14*I + 34, -42*I - 522, 8*I + 12]])
sage: sv = A.singular_values()
sage: sv[0:3]
[1440.733666, 18.4044034134, 6.83970779714]
sage: (10^-15 < sv[3]) and (sv[3] < 10^-13)
True
sage: (10^-16 < sv[4]) and (sv[4] < 10^-14)
True
A full-rank matrix that is ill-conditioned. We use this to illustrate ways of using the various possibilities for eps, including one that is ill-advised. Notice that the automatically computed cutoff gets this (difficult) example slightly wrong. This illustrates the impossibility of any automated process always getting this right. Use with caution and judgement.
sage: entries = [1/(i+j+1) for i in range(12) for j in range(12)]
sage: B = matrix(QQ, 12, 12, entries)
sage: B.rank()
12
sage: A = B.change_ring(RDF)
sage: A.condition() > 1.6e16 or A.condition()
True
sage: A.singular_values(eps=None) # abs tol 1e-16
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 1.11633574832e-05, 4.08237611034e-07, 1.12286106622e-08, 2.25196451406e-10, 3.11134627616e-12, 2.65006299814e-14, 1.00050293715e-16]
sage: A.singular_values(eps='auto') # abs tol 1e-16
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 1.11633574832e-05, 4.08237611034e-07, 1.12286106622e-08, 2.25196451406e-10, 3.11134627616e-12, 2.65006299814e-14, 0.0]
sage: A.singular_values(eps=1e-4)
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
With Sage’s “verbose” facility, you can compactly see the cutoff at work. In any application of this routine, or those that build upon it, it would be a good idea to conduct this exercise on samples. We also test here that all the values are returned in \(RDF\) since singular values are always real.
sage: A = matrix(CDF, 4, range(16))
sage: set_verbose(1)
sage: sv = A.singular_values(eps='auto'); sv
verbose 1 (<module>) singular values,
smallest-non-zero:cutoff:largest-zero,
2.2766...:6.2421...e-14:...
[35.139963659, 2.27661020871, 0.0, 0.0]
sage: set_verbose(0)
sage: all([s in RDF for s in sv])
True
TESTS:
Bogus values of the eps keyword will be caught.
sage: A.singular_values(eps='junk')
Traceback (most recent call last):
...
ValueError: could not convert string to float: junk
REFERENCES:
[WATKINS] | Watkins, David S. Fundamentals of Matrix Computations, Third Edition. Wiley, Hoboken, New Jersey, 2010. |
AUTHOR:
Solve the vector equation x*A = b for a nonsingular A.
INPUT:
OUTPUT:
The unique solution x to the matrix equation x*A = b, as a vector over the same base ring as self.
ALGORITHM:
Uses the solve() routine from the SciPy scipy.linalg module, after taking the tranpose of the coefficient matrix.
EXAMPLES:
Over the reals.
sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
[ 1.0 2.0 5.0]
[ 7.6 2.3 1.0]
[ 1.0 2.0 -1.0]
sage: b = vector(RDF,[1,2,3])
sage: x = A.solve_left(b); x.zero_at(1e-18) # fix noisy zeroes
(0.666666666..., 0.0, 0.333333333...)
sage: x.parent()
Vector space of dimension 3 over Real Double Field
sage: x*A
(1.0, 2.0, 3.0)
Over the complex numbers.
sage: A = matrix(CDF, [[ 0, -1 + 2*I, 1 - 3*I, I],
... [2 + 4*I, -2 + 3*I, -1 + 2*I, -1 - I],
... [ 2 + I, 1 - I, -1, 5],
... [ 3*I, -1 - I, -1 + I, -3 + I]])
sage: b = vector(CDF, [2 -3*I, 3, -2 + 3*I, 8])
sage: x = A.solve_left(b); x
(-1.55765124... - 0.644483985...*I, 0.183274021... + 0.286476868...*I, 0.270818505... + 0.246619217...*I, -1.69003558... - 0.828113879...*I)
sage: x.parent()
Vector space of dimension 4 over Complex Double Field
sage: abs(x*A - b) < 1e-14
True
The vector of constants, b, can be given in a variety of forms, so long as it coerces to a vector over the same base ring as the coefficient matrix.
sage: A=matrix(CDF, 5, [1/(i+j+1) for i in range(5) for j in range(5)])
sage: A.solve_left([1]*5) # rel tol 1e-11
(5.0, -120.0, 630.0, -1120.0, 630.0)
TESTS:
A degenerate case.
sage: A = matrix(RDF, 0, 0, [])
sage: A.solve_left(vector(RDF,[]))
()
The coefficent matrix must be square.
sage: A = matrix(RDF, 2, 3, range(6))
sage: b = vector(RDF, [1,2,3])
sage: A.solve_left(b)
Traceback (most recent call last):
...
ValueError: coefficient matrix of a system over RDF/CDF must be square, not 2 x 3
The coefficient matrix must be nonsingular.
sage: A = matrix(RDF, 5, range(25))
sage: b = vector(RDF, [1,2,3,4,5])
sage: A.solve_left(b)
Traceback (most recent call last):
...
LinAlgError: singular matrix
The vector of constants needs the correct degree.
sage: A = matrix(RDF, 5, range(25))
sage: b = vector(RDF, [1,2,3,4])
sage: A.solve_left(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Real Double Field incompatible with matrix over Real Double Field
The vector of constants needs to be compatible with the base ring of the coefficient matrix.
sage: F.<a> = FiniteField(27)
sage: b = vector(F, [a,a,a,a,a])
sage: A.solve_left(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Finite Field in a of size 3^3 incompatible with matrix over Real Double Field
With a coefficient matrix over RDF, a vector of constants over CDF can be accomodated by converting the base ring of the coefficient matrix.
sage: A = matrix(RDF, 2, range(4))
sage: b = vector(CDF, [1+I,2])
sage: A.solve_left(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Complex Double Field incompatible with matrix over Real Double Field
sage: B = A.change_ring(CDF)
sage: B.solve_left(b)
(0.5 - 1.5*I, 0.5 + 0.5*I)
Solve the equation \(A x = b\) using LU decomposition.
Warning
This function is broken. See trac 4932.
INPUT:
Note
This method precomputes and stores the LU decomposition before solving. If many equations of the form Ax=b need to be solved for a singe matrix A, then this method should be used instead of solve. The first time this method is called it will compute the LU decomposition. If the matrix has not changed then subsequent calls will be very fast as the precomputed LU decomposition will be reused.
EXAMPLES:
sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
[ 1.0 2.0 5.0]
[ 7.6 2.3 1.0]
[ 1.0 2.0 -1.0]
sage: b = vector(RDF,[1,2,3])
sage: x = A.solve_left_LU(b); x
Traceback (most recent call last):
...
NotImplementedError: this function is not finished (see trac 4932)
TESTS:
We test two degenerate cases:
sage: A = matrix(RDF, 0, 3, [])
sage: A.solve_left_LU(vector(RDF,[]))
(0.0, 0.0, 0.0)
sage: A = matrix(RDF, 3, 0, [])
sage: A.solve_left_LU(vector(RDF,3, [1,2,3]))
()
Solve the vector equation A*x = b for a nonsingular A.
INPUT:
OUTPUT:
The unique solution x to the matrix equation A*x = b, as a vector over the same base ring as self.
ALGORITHM:
Uses the solve() routine from the SciPy scipy.linalg module.
EXAMPLES:
Over the reals.
sage: A = matrix(RDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]); A
[ 1.0 2.0 5.0]
[ 7.6 2.3 1.0]
[ 1.0 2.0 -1.0]
sage: b = vector(RDF,[1,2,3])
sage: x = A.solve_right(b); x
(-0.113695090439, 1.39018087855, -0.333333333333)
sage: x.parent()
Vector space of dimension 3 over Real Double Field
sage: A*x
(1.0, 2.0, 3.0)
Over the complex numbers.
sage: A = matrix(CDF, [[ 0, -1 + 2*I, 1 - 3*I, I],
... [2 + 4*I, -2 + 3*I, -1 + 2*I, -1 - I],
... [ 2 + I, 1 - I, -1, 5],
... [ 3*I, -1 - I, -1 + I, -3 + I]])
sage: b = vector(CDF, [2 -3*I, 3, -2 + 3*I, 8])
sage: x = A.solve_right(b); x
(1.96841637... - 1.07606761...*I, -0.614323843... + 1.68416370...*I, 0.0733985765... + 1.73487544...*I, -1.6018683... + 0.524021352...*I)
sage: x.parent()
Vector space of dimension 4 over Complex Double Field
sage: abs(A*x - b) < 1e-14
True
The vector of constants, b, can be given in a variety of forms, so long as it coerces to a vector over the same base ring as the coefficient matrix.
sage: A=matrix(CDF, 5, [1/(i+j+1) for i in range(5) for j in range(5)])
sage: A.solve_right([1]*5) # rel tol 1e-11
(5.0, -120.0, 630.0, -1120.0, 630.0)
TESTS:
A degenerate case.
sage: A = matrix(RDF, 0, 0, [])
sage: A.solve_right(vector(RDF,[]))
()
The coefficent matrix must be square.
sage: A = matrix(RDF, 2, 3, range(6))
sage: b = vector(RDF, [1,2,3])
sage: A.solve_right(b)
Traceback (most recent call last):
...
ValueError: coefficient matrix of a system over RDF/CDF must be square, not 2 x 3
The coefficient matrix must be nonsingular.
sage: A = matrix(RDF, 5, range(25))
sage: b = vector(RDF, [1,2,3,4,5])
sage: A.solve_right(b)
Traceback (most recent call last):
...
LinAlgError: singular matrix
The vector of constants needs the correct degree.
sage: A = matrix(RDF, 5, range(25))
sage: b = vector(RDF, [1,2,3,4])
sage: A.solve_right(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Real Double Field incompatible with matrix over Real Double Field
The vector of constants needs to be compatible with the base ring of the coefficient matrix.
sage: F.<a> = FiniteField(27)
sage: b = vector(F, [a,a,a,a,a])
sage: A.solve_right(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Finite Field in a of size 3^3 incompatible with matrix over Real Double Field
With a coefficient matrix over RDF, a vector of constants over CDF can be accomodated by converting the base ring of the coefficient matrix.
sage: A = matrix(RDF, 2, range(4))
sage: b = vector(CDF, [1+I,2])
sage: A.solve_right(b)
Traceback (most recent call last):
...
TypeError: vector of constants over Complex Double Field incompatible with matrix over Real Double Field
sage: B = A.change_ring(CDF)
sage: B.solve_right(b)
(-0.5 - 1.5*I, 1.0 + 1.0*I)
Return the transpose of this matrix, without changing self.
EXAMPLES:
sage: m = matrix(RDF,2,3,range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: m2 = m.transpose()
sage: m[0,0] = 2
sage: m2 #note that m2 hasn't changed
[0.0 3.0]
[1.0 4.0]
[2.0 5.0]
.T is a convenient shortcut for the transpose:
sage: m.T
[2.0 3.0]
[1.0 4.0]
[2.0 5.0]
sage: m = matrix(RDF,0,3); m
[]
sage: m.transpose()
[]
sage: m.transpose().parent()
Full MatrixSpace of 3 by 0 dense matrices over Real Double Field
Returns a copy of the matrix where elements smaller than or equal to eps are replaced with zeroes. For complex matrices, the real and imaginary parts are considered individually.
This is useful for modifying output from algorithms which have large relative errors when producing zero elements, e.g. to create reliable doctests.
INPUT:
OUTPUT:
A modified copy of the matrix.
EXAMPLES:
sage: a=matrix([[1, 1e-4r, 1+1e-100jr], [1e-8+3j, 0, 1e-58r]])
sage: a
[ 1.0 0.0001 1.0 + 1e-100*I]
[ 1e-08 + 3.0*I 0.0 1e-58]
sage: a.zero_at(1e-50)
[ 1.0 0.0001 1.0]
[1e-08 + 3.0*I 0.0 0.0]
sage: a.zero_at(1e-4)
[ 1.0 0.0 1.0]
[3.0*I 0.0 0.0]