How I can separate a matrix that is not positive definite, into two matrices?

I want to separate a matrix that is not positive definite, into two matrices in Matlab like this:

for example this matrix:

Answer 1

[Sam Chak]

Hi @hossen hassanzadth

Your expression "" appears to resemble an attempt at performing a Cholesky decomposition. However, it's important to note that a Cholesky decomposition is defined for symmetric positive-definite matrices. In the case of a symmetric indefinite matrix, a Cholesky decomposition does not exist by definition.

Nevertheless, you can explore an alternative approach by seeking a decomposition in the form of , where represents a diagonal matrix with entries that can be both positive and negative.

Q  = [-92.316   31.78  240.417;
       31.78  -194.66  275.47;
      240.417  275.47  938.99];
E  = eig(Q)     % check if indefinite matrix

E = 3×1

1.0e+03 * -0.2623 -0.1390 1.0533

[L, D] = ldl(Q)     % L*D*L' factorization

L = 3×3

0.2560 0.1407 1.0000 0.2934 1.0000 0 1.0000 0 0

D = 3×3

938.9900 0 0 0 -275.4742 0 0 0 -148.4208

Sq = L'

Sq = 3×3

0.2560 0.2934 1.0000 0.1407 1.0000 0 1.0000 0 0

resnorm = norm(Q - Sq'*D*Sq, 'fro')/norm(Q, 'fro')

resnorm = 2.5972e-17

Sq'*D*Sq

ans = 3×3

-92.3160 31.7800 240.4170 31.7800 -194.6600 275.4700 240.4170 275.4700 938.9900

Answer 2

[John D'Errico]

Please learn to use operators and to clearly explain your question.

Are you asking to find a new matrix Sq, such that the linear algebraic product Sq'*sq is equal to Q, where Q is NOT positive definite? NO. That is impossible.

Are you asking to find two matrices S and q, such that the product of the 4 matrices S*q'*S*q is Q? (I highly doubt this is your question, but you explicitly said TWO matrices.)

Since the first is impossible, you asking to find some matrix Sq such that Sq' * Sq is as close as possible to Q, based on some norm on the difference?

Are your matrices real, or are they complex? Must the solution live in the real domain?

[hossen hassanzadth]

hi

I want to find Sq, such that the linear algebraic product Sq'*Sq is equal or as close as possible to Q.

solution can be in real or complex domain.