MATLAB: Creat a positive definite matrix, which way is better

MATLABnumerical analysisnumerical stabilitypositive definite matrix; numerical error;

Hi everyone,

Suppose I have a positive definite matrix P and a non-singular matrix A, I realised there is a small numerical difference between K=(I-A)*P*(I-A)' and K_=P-A*P-P*A'+A*P*A'. If I want to create a positive definite matrix, which way should I use?

In some extreme cases, is it possible that K or K_ has very small negative eigenvalues?

Thanks

Best Answer

Yes, if A has eigenvalues close to 1, then I believe neither K nor K_ will be positive definite with numerical robustness.

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MATLAB: Smallest non-zero eigenvalue for a generalized eigenvalue problem

Since both matrices A and B are singular, it is not an easy problem numerically. Even eig(full(A), full(B)) may give you wrong answers. EIGS in the call eigs(A, B, k, guess) is not probably going to work, since it relies on computing the inverse of the matrix A - guess B, which is singular in your case.

If the kernel of B is a subset of the kernel of A, and the difference in dimensions is not too large, you may want to try https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m with the constraint blockVectorY being the set of vectors spanning the known kernel of B. LOBPCG gives you k smallest, no matter zeros, or not, however, so you may need to choose k large enough to get to where you need. The value of k in LOBPCG is the number of the linearly independent initial approximate eigenvectors. If k is too large, relative to the matrix size, it will not work well...

MATLAB: How to determine if a matrix is positive definite using MATLAB

This change has been incorporated into the documentation in Release 14 Service Pack 3 (R14SP3). For previous releases, read below for any additional information:

Rather than using the EIG function to obtain the eigenvalues in order to determine positive definiteness, it is more computationally efficient to use the CHOL function. The CHOL function provides an optional second output argument "p" which is zero if the matrix is found to be positive definite. If the input matrix is not positive definite, then "p" will be a positive integer:

 >>[~,p] = chol(zeros(3))
 p =
        1

The CHOL function will return an error if it is only provided with a single output argument, and is also given a matrix that is not positive definite. NOTE: CHOL expects its input matrix to be symmetric and only looks at the upper triangular portion of the matrix.

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MATLAB: Smallest non-zero eigenvalue for a generalized eigenvalue problem

MATLAB: How to determine if a matrix is positive definite using MATLAB

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