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: ( L*L.transpose() - m ).norm(1) < 2^-50 True Here is a Hermitian example: sage: r = matrix(CDF, 2, 2, [ 1, -2*I, 2*I, 6 ]); r [ 1.0 -2.0*I] [ 2.0*I 6.0] sage: r.eigenvalues() [0.298437881284, 6.70156211872] sage: ( r - r.conjugate().transpose() ).norm(1) < 1e-30 True sage: L = r.cholesky_decomposition(); L [ 1.0 0.0] [ 2.0*I 1.41421356237] sage: ( r - L*L.conjugate().transpose() ).norm(1) < 1e-30 True sage: L.parent() Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field TESTS: The following examples are not positive definite: sage: m = -identity_matrix(3).change_ring(RDF) sage: m.cholesky_decomposition() Traceback (most recent call last): ... ValueError: The input matrix was not symmetric and positive definite sage: m = -identity_matrix(2).change_ring(RealField(100)) sage: m.cholesky_decomposition() Traceback (most recent call last): ... ValueError: The input matrix was not symmetric and positive definite Here is a large example over a higher precision complex field: [12.1121838768, 5.17714373118, 0.183583821657, 0.798520682956, 2.03399202232, 3.81092266261] sage: r = MatrixSpace(ComplexField(100), 6, 6).random_element() sage: m = r * r.conjugate().transpose() sage: m.change_ring(CDF) # for display purposes [ 4.03491289478 1.65865229397 + 1.20395241554*I -0.275377464753 - 0.3 [ 1.65865229397 - 1.20395241554*I 5.19463366777 0.192784646673 + 0.2 [-0.275377464753 + 0.392579363912*I 0.192784646673 - 0.217539084881*I [ 0.646019176609 + 1.80427378747*I -1.24630239913 + 1.00510523556*I -0.550558880479 - 0.3 [ 1.15898675468 - 2.35202344518*I 1.65179716714 - 1.27031304403*I 1.00862850855 - 0.9 [ -1.07920143474 - 1.37815737417*I 1.17275462994 - 0.565615358757*I -0.740344951784 + 0. sage: eigs = m.change_ring(CDF).eigenvalues() # again for display purposes sage: all(abs(imag(e)) < 1.3e-15 for e in eigs) True sage: [real(e) for e in eigs] sage: ( m - m.conjugate().transpose() ).norm(1) < 1e-50 True sage: L = m.cholesky_decomposition(); L.change_ring(CDF) [ 2.00870926089 0.0 [ 0.825730396261 - 0.599366189511*I 2.03802923221 [ -0.137091748475 + 0.195438618996*I 0.20761467212 - 0.145606613292*I [ 0.321609099528 + 0.898225453828*I -0.477666770113 + 0.0346666053769*I -0.416429223553 - 140 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 0.576980839012 - 1.17091282993*I 0.232362216253 - 0.318581071175*I 0.880672963687 - [ -0.537261143636 - 0.686091014267*I 0.591339766401 + 0.158450627525*I -0.561877938537 + sage: ( m - L*L.conjugate().transpose() ).norm(1) < 1e-20 True sage: L.parent() Full MatrixSpace of 6 by 6 dense matrices over Complex Field with 100 bits of precision 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 Here is an example that returns an incorrect answer, because the input is not positive definite: sage: r = matrix(CDF, 2, 2, [ 1, -2*I, 2*I, 0 ]); r [ 1.0 -2.0*I] [ 2.0*I 0.0] sage: r.eigenvalues() [2.56155281281, -1.56155281281] sage: ( r - r.conjugate().transpose() ).norm(1) < 1e-30 True sage: L = r.cholesky_decomposition(); L [ 1.0 0.0] [2.0*I 2.0*I] sage: L*L.conjugate().transpose() [ 1.0 -2.0*I] [ 2.0*I 8.0] This test verifies that the caching of the two variants of the Cholesky decomposition have been cleanly separated. It can be safely removed as part of removing this method at the end of the deprecation period. (From trac ticket #13045.) sage: r = matrix(CDF, 2, 2, [ 0, -2*I, 2*I, 0 ]); r [ 0.0 -2.0*I] [ 2.0*I 0.0] sage: r.cholesky_decomposition() [ 0.0 0.0] [NaN + NaN*I NaN + NaN*I] sage: r.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite sage: r[0,0] = 0 # clears cache sage: r.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite sage: r.cholesky_decomposition() [ 0.0 0.0] [NaN + NaN*I NaN + NaN*I] column_module() Return the free module over the base ring spanned by the columns of this matrix. EXAMPLES: sage: t = matrix(QQ, 3, range(9)); t [0 1 2] [3 4 5] [6 7 8] 141
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