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: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I], ... [4 + 2*I, 8, 10*I], ... [2 - 2*I, -10*I, -3]]) sage: B.is_hermitian() True sage: [ev.real() for ev in B.eigenvalues()] [15.88..., 0.08..., -8.97...] sage: B.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite TESTS: A trivial case. sage: A = matrix(RDF, 0, []) sage: A.cholesky() [] The Cholesky factorization is only defined for square matrices. sage: A = matrix(RDF, 4, 5, range(20)) sage: A.cholesky() Traceback (most recent call last): ... ValueError: Cholesky decomposition requires a square matrix, not a 4 x 5 matrix condition(p=’frob’) Returns the condition number of a square nonsingular matrix. Roughly speaking, this is a measure of how sensitive the matrix is to round-off errors in numerical computations. The minimum possible value is 1.0, and larger numbers indicate greater sensitivity. INPUT: •p - default: ‘frob’ - controls which norm is used to compute the condition number, allowable values are ‘frob’ (for the Frobenius norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics. OUTPUT: The condition number of a matrix is the product of a norm of the matrix times the norm of the inverse of the matrix. This requires that the matrix be square and invertible (nonsingular, full rank). Returned value is a double precision floating point value in RDF, or Infinity. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number a + bi is √ a 2 + b 2 . Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition. •p = ’frob’: the default norm employed in computing the condition number, the Frobenius norm, which for a matrix A = (a ij ) computes ⎛ ⎝ ∑ i,j ⎞1/2 |a i,j | 2 ⎠ •p = ’sv’: the quotient of the maximal and minimal singular value. •p = Infinity or p = oo: the maximum row sum. 358 Chapter 19. Dense matrices using a NumPy backend.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 •p = -Infinity or p = -oo: the minimum column sum. •p = 1: the maximum column sum. •p = -1: the minimum column sum. •p = 2: the 2-norm, equal to the maximum singular value. •p = -2: the minimum singular value. ALGORITHM: Computation is performed by the cond() function of the SciPy/NumPy library. EXAMPLES: First over the reals. sage: A = matrix(RDF, 4, [(1/4)*x^3 for x in range(16)]); A [ 0.0 0.25 2.0 6.75] [ 16.0 31.25 54.0 85.75] [ 128.0 182.25 250.0 332.75] [ 432.0 549.25 686.0 843.75] sage: A.condition() 9923.88955... sage: A.condition(p=’frob’) 9923.88955... sage: A.condition(p=Infinity) 22738.5 sage: A.condition(p=-Infinity) 17.5 sage: A.condition(p=1) 12139.21... sage: A.condition(p=-1) 550.0 sage: A.condition(p=2) 9897.8088... sage: A.condition(p=-2) 0.000101032462... And over the complex numbers. sage: B = matrix(CDF, 3, [x + x^2*I for x in range(9)]); B [ 0.0 1.0 + 1.0*I 2.0 + 4.0*I] [ 3.0 + 9.0*I 4.0 + 16.0*I 5.0 + 25.0*I] [6.0 + 36.0*I 7.0 + 49.0*I 8.0 + 64.0*I] sage: B.condition() 203.851798... sage: B.condition(p=’frob’) 203.851798... sage: B.condition(p=Infinity) 369.55630... sage: B.condition(p=-Infinity) 5.46112969... sage: B.condition(p=1) 289.251481... sage: B.condition(p=-1) 20.4566639... sage: B.condition(p=2) 202.653543... sage: B.condition(p=-2) 0.00493453005... 359
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