How I can separate a matrix that is not positive definite, into two matrices?
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I want to separate a matrix that is not positive definite, into two matrices in Matlab like this:
for example this matrix:
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Accepted Answer
Sam Chak
on 13 Oct 2023
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
[L, D] = ldl(Q) % L*D*L' factorization
Sq = L'
resnorm = norm(Q - Sq'*D*Sq, 'fro')/norm(Q, 'fro')
Sq'*D*Sq
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More Answers (1)
John D'Errico
on 31 Mar 2021
Edited: John D'Errico
on 31 Mar 2021
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?
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