Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: image(A)
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1 0 0 426]
[ 0 1 0 518]
[ 0 0 1 293]
[ 0 0 0 687]
sage: image(B) == B.row_module()
True
indefinite_factorization(algorithm=’symmetric’, check=True)
Decomposes a symmetric or Hermitian matrix into a lower triangular matrix and a diagonal matrix.
INPUT:
•self - a square matrix over a ring. The base ring must have an implemented fraction field.
•algorithm - default: ’symmetric’. Either ’symmetric’ or ’hermitian’, according to if
the input matrix is symmetric or hermitian.
•check - default: True - if True then performs the check that the matrix is consistent with the
algorithm keyword.
OUTPUT:
A lower triangular matrix L with each diagonal element equal to 1 and a vector of entries that form a
diagonal matrix D. The vector of diagonal entries can be easily used to form the matrix, as demonstrated
below in the examples.
For a symmetric matrix, A, these will be related by
A = LDL T
If A is Hermitian matrix, then the transpose of L should be replaced by the conjugate-transpose of L.
If any leading principal submatrix (a square submatrix in the upper-left corner) is singular then this method
will fail with a ValueError.
ALGORITHM:
The algorithm employed only uses field operations, but the compuation of each diagonal entry has the
potential for division by zero. The number of operations is of order n 3 /3, which is half the count for
an LU decomposition. This makes it an appropriate candidate for solving systems with symmetric (or
Hermitian) coefficient matrices.
EXAMPLES:
There is no requirement that a matrix be positive definite, as indicated by the negative entries in the resulting
diagonal matrix. The default is that the input matrix is symmetric.
sage: A = matrix(QQ, [[ 3, -6, 9, 6, -9],
... [-6, 11, -16, -11, 17],
... [ 9, -16, 28, 16, -40],
... [ 6, -11, 16, 9, -19],
... [-9, 17, -40, -19, 68]])
sage: A.is_symmetric()
True
sage: L, d = A.indefinite_factorization()
sage: D = diagonal_matrix(d)
sage: L
[ 1 0 0 0 0]
181