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 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 sage: A.eigenvalues() [30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0 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 2×2 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.. sage: sorted(A.eigenvalues()) [-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336.. sage: abs(A.norm()-T.norm()) < 1e-12 True Starting with complex numbers and requesting a result over the reals will never happen. 384 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 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], 385
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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