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 Some subfields of the complex numbers, such as this number field of complex numbers with rational real and imaginary parts, allow for this computation. sage: C. = QuadraticField(-1) sage: A = matrix(C, [[ 23, 17*I + 3, 24*I + 25, 21*I], ... [ -17*I + 3, 38, -69*I + 89, 7*I + 15], ... [-24*I + 25, 69*I + 89, 976, 24*I + 6], ... [ -21*I, -7*I + 15, -24*I + 6, 28]]) sage: A.is_hermitian() True sage: L = A.cholesky() sage: L [ 4.79... 0 0 0] [ 0.62... - 3.54...*I 5.00... 0 0] [ 5.21... - 5.00...*I 13.58... + 10.72...*I 24.98... 0] [ -4.37...*I -0.10... - 0.85...*I -0.21... + 0.37...*I 2.81...] sage: L.parent() Full MatrixSpace of 4 by 4 dense matrices over Algebraic Field sage: (L*L.conjugate_transpose() - A.change_ring(QQbar)).norm() < 10^-10 True The field of algebraic numbers is an ideal setting for this computation. sage: A = matrix(QQbar, [[ 2, 4 + 2*I, 6 - 4*I], ... [ -2*I + 4, 11, 10 - 12*I], ... [ 4*I + 6, 10 + 12*I, 37]]) sage: A.is_hermitian() True sage: L = A.cholesky() sage: L [ 1.414213562373095 0 0] [2.828427124746190 - 1.414213562373095*I 1 0] [4.242640687119285 + 2.828427124746190*I -2*I + 2 1.732050807568878] sage: L.parent() Full MatrixSpace of 3 by 3 dense matrices over Algebraic Field sage: (L*L.conjugate_transpose() - A.change_ring(QQbar)).norm() < 10^-10 True Results are cached, hence immutable. Use the copy function if you need to make a change. sage: A = matrix(QQ, [[ 4, -2, 4, 2], ... [-2, 10, -2, -7], ... [ 4, -2, 8, 4], ... [ 2, -7, 4, 7]]) sage: L = A.cholesky() sage: L.is_immutable() True sage: from copy import copy sage: LC = copy(L) sage: LC[0,0] = 1000 sage: LC [1000 0 0 0] [ -1 3 0 0] [ 2 0 2 0] [ 1 -2 1 1] There are a variety of situations which will prevent the computation of a Cholesky decomposition. The base ring must be exact. For numerical work, create a matrix with a base ring of RDF or CDF and use 136 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 the cholesky() method for matrices of that type. sage: F = RealField(100) sage: A = matrix(F, [[1.0, 3.0], [3.0, -6.0]]) sage: A.cholesky() Traceback (most recent call last): ... TypeError: base ring of the matrix must be exact, not Real Field with 100 bits of precision The base ring may not have a fraction field. sage: A = matrix(Integers(6), [[2, 0], [0, 4]]) sage: A.cholesky() Traceback (most recent call last): ... ValueError: unable to check positive definiteness because Unable to create the fraction field of Ring of integers modulo 6 The base field may not have elements that are comparable to zero. sage: F. = FiniteField(5^4) sage: A = matrix(F, [[2+a^3, 3], [3, 3]]) sage: A.cholesky() Traceback (most recent call last): ... ValueError: unable to check positive definiteness because cannot convert computations from Finite Field in a of size 5^4 into real numbers The algebraic closure of the fraction field of the base ring may not be implemented. sage: F = Integers(7) sage: A = matrix(F, [[4, 0], [0, 3]]) sage: A.cholesky() Traceback (most recent call last): ... TypeError: base field needs an algebraic closure with square roots, not Ring of integers modulo 7 The matrix may not be positive definite. sage: C. = QuadraticField(-1) sage: B = matrix(C, [[ 2, 4 - 2*I, 2 + 2*I], ... [4 + 2*I, 8, 10*I], ... [2 - 2*I, -10*I, -3]]) sage: B.is_positive_definite() False sage: B.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite, so cannot compute Cholesky decomposition The matrix could be positive semi-definite, and thus lack a Cholesky decomposition. sage: A = matrix(QQ, [[21, 15, 12, -3], ... [15, 12, 9, 12], ... [12, 9, 7, 3], ... [-3, 12, 3, 8]]) sage: A.is_positive_definite() False 137
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