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 ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a LU_valid() 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 QR() 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 A = QR, Q ∗ Q = I where R is upper-triangular. Q ∗ 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) 352 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 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() 353
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Sage Reference Manual: Matrices and
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