Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[-2 1 0 0 0]
[ 3 -2 1 0 0]
[ 2 -1 0 1 0]
[-3 1 -3 1 1]
sage: D
[ 3 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 5 0 0]
[ 0 0 0 -2 0]
[ 0 0 0 0 -1]
sage: A == L*D*L.transpose()
True
Optionally, Hermitian matrices can be factored and the result has a similar property (but not identical).
Here, the field is all complex numbers with rational real and imaginary parts. As theory predicts, the
diagonal entries will be real numbers.
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_hermitian()
True
sage: L, d = B.indefinite_factorization(algorithm=’hermitian’)
sage: D = diagonal_matrix(d)
sage: L
[ 1 0 0]
[ I + 2 1 0]
[ -I + 1 2*I + 1 1]
sage: D
[ 2 0 0]
[ 0 -2 0]
[ 0 0 3]
sage: B == L*D*L.conjugate_transpose()
True
If a leading principal submatrix has zero determinant, this algorithm will fail. This will never happen with
a positive definite matrix.
sage: A = matrix(QQ, [[21, 15, 12, -2],
... [15, 12, 9, 6],
... [12, 9, 7, 3],
... [-2, 6, 3, 8]])
sage: A.is_symmetric()
True
sage: A[0:3,0:3].det() == 0
True
sage: A.indefinite_factorization()
Traceback (most recent call last):
...
ValueError: 3x3 leading principal submatrix is singular,
so cannot create indefinite factorization
This algorithm only depends on field operations, so outside of the singular submatrix situation, any matrix
may be factored. This provides a reasonable alternative to the Cholesky decomposition.
sage: F. = FiniteField(5^3)
sage: A = matrix(F,
... [[ a^2 + 2*a, 4*a^2 + 3*a + 4, 3*a^2 + a, 2*a^2 + 2*a + 1],
182 Chapter 7. Base class for matrices, part 2