Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 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 368 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: B.is_hermitian(algorithm=’naive’, tol=1.0e-17) False sage: B.is_hermitian(algorithm=’naive’, tol=1.0e-15) True # random 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: •Rob Beezer (2011-03-30) is_normal(tol=1e-12, algorithm=’orthonormal’) Returns True if the matrix commutes with its conjugate-transpose. INPUT: •tol - default: 1e-12 - the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal. •algorithm - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure. 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: 369
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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