Singularity of Positive Semi Definite Matrices

[fawaz1]

Hello,I'm given a sparse symmetric positive semi definite matrix and I want to check whether it is singular or not.What's a quick way to do that? (I can't do any kind of factorization because the matrix can be huge)I know the following:- if any of the diagonal entries is zero, then the matrix is singular (because then I have a row and a column of zeros)- If I multiply the matrix by a vector of ones and I get zeros everywhere, then the matrix is singular.What other cases am I missing?Thank youMohammad

 

[jvicens]

Hi Mohammad,

There are a few other cases you could consider when checking for singularity in a sparse symmetric positive semi-definite matrix. Here are a few suggestions:

1. Check for any zero rows or columns: If a row or column contains all zeros, then the matrix is singular.

2. Look for linearly dependent rows or columns: If two or more rows or columns are linearly dependent, then the matrix is singular.

3. Use the rank-deficiency theorem: If the rank of the matrix is less than the number of rows or columns, then the matrix is singular.

4. Check for zero eigenvalues: If any of the eigenvalues of the matrix are zero, then the matrix is singular.

5. Utilize the Gershgorin circle theorem: This theorem states that the eigenvalues of a matrix are contained within the union of the Gershgorin circles, which are defined by the diagonal entries and the sum of the absolute values of the off-diagonal entries in each row. If all the Gershgorin circles are contained within the origin, then the matrix is singular.

I hope these suggestions help you in quickly determining singularity in your matrix. Good luck with your research!