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 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 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] SVD(*args, **kwds) 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. 354 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 INPUT: •A – a matrix OUTPUT: •U, S, V – immutable matrices such that A = U ∗ S ∗ V.conj().transpose() where U and V are orthogonal and S is zero off of the diagonal. 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] 355
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