Why doesnt chol work for hermitian positive semi-definite matrix?

Hello,

I am trying to perform factorization of form A = R'*R using matlab function 'chol' but A is not positive definite, rather its hermitian positive semi-definite. As per theory from https://en.wikipedia.org/wiki/Hermitian_matrix , I should be able to perform factorization but matlab expects A to be positive definite matrix. Is there an alternative function in matlab?

Thanks,

[Geoff Hayes]

@Nikhil Challa - I don't think it is MATLAB imposing the condition that the input matrix must be positive definite but rather this is what the Cholesky decomposition requires (see https://en.wikipedia.org/wiki/Cholesky_decomposition). In your case, you may be able to use the Eignevalue decomposition. (I say "may be" only because it has been a long time since I've given any thought to this.)

[Nikhil Challa]

@Geoff Hayes , the https://en.wikipedia.org/wiki/Cholesky_decomposition link also talks about decomposition for hermitian positive semi-definite matrix except the solution may not be unique. Matlab has lot of functions that provide solution even for non-unique cases, so I would expect matlab to do the same for 'chol' function also.

Let me know if my understanding is incorrect.

[Torsten]

https://de.mathworks.com/matlabcentral/answers/477584-problem-with-cholesky-s-decomposition-of-a-positive-semi-definite-matrix

Answer 1

[Pravarthana P]

Hi Nikil Challa,

I understand that you are trying to perform factorization using “chol” function and are facing an issue with positive semi-definite matrix.

The “chol” function expects a symmetric positive definite matrix as input as mentioned in the documentation link.

The following can likely be a workaround to factorize positive semi-definite matrix:

1. Compute the LDL decomposition: A lower-triangular matrix L, a block-diagonal matrix D (1-by-1 and 2-by-2 blocks), and a permutation matrix P, such that A is P*L*D*L'*P' :

[L, D, P] = ldl(A);

2.Compute the eigenvalue decomposition, set its negative eigenvalues to zero, and use QR to get two triangular factors for this modified matrix:

[U, d] = eig(A, 'vector'); % A == U*diag(d)*U'
d(d < 0) = 0;  % Set negative eigenvalues of A to zero
[~, R] = qr(diag(sqrt(d))*U');  
% Q*R == sqrt(D)*U', so A == U*diag(d)*U'== R'*Q'*Q*R == R'*R
% In this case, check that d(d<0) are all nearly zero.

3.Add a small number to A’s diagonal before calling Cholesky:

 R=chol(A+1e-13*eye(size(A)));

To know more information on why the “chol” function cannot be used to factorize positive semi-definite matrix, you can refer to the link.

I hope this information helps you.