# PSDdecomposition -- factorization of a positive semidefinite matrix

## Synopsis

• Usage:
(L,D,P) = PSDdecomposition A
• Inputs:
• A, ,
• Outputs:
• L, , nonsingular
• D, , diagonal
• P, , permutation matrix

## Description

Given a positive semidefinite matrix $A$, this method factorizes it in the form $P' A P = L D L'$, where $P$ is a permutation matrix, $L$ is nonsingular, $D$ is diagonal. If $A$ is a real matrix, this factorization is obtained from its eigenvalue decomposition. For rational matrices we use the LDL decomposition [Golub-vanLoan'89]. The method returns null if $A$ is not positive semidefinite.

 i1 : A = matrix(QQ, {{5,3,5},{3,2,4},{5,4,10}}) o1 = | 5 3 5 | | 3 2 4 | | 5 4 10 | 3 3 o1 : Matrix QQ <--- QQ i2 : (L,D,P) = PSDdecomposition(A) o2 = (| 1 0 0 |, | 10 0 0 |, | 0 1 0 |) | 1/2 1 0 | | 0 5/2 0 | | 0 0 1 | | 2/5 2/5 1 | | 0 0 0 | | 1 0 0 | o2 : Sequence i3 : L*D*transpose(L) == transpose(P)*A*P o3 = true

References: Matrix Computations, Gene Golub and Charles van Loan. Johns Hopkins series in the Mathematical Science (1989), 2 ed., pp. 133-148.

## For the programmer

The object PSDdecomposition is .