I heard that
for a Hermitian or symmetric matrix, if it is positive semi-definite, then all its square submatrices (not just those along diagonal) are non-negative
I suspect the statement is wrong.
There can be different interpretations of "non-negative".
If the statement meant to say "all its square submatrices (not just those along diagonal) have non-negative determinants", then matrix $[2,-1;-1,2]$ would be a positive semidefinite matrix but the square submatrix $[-1]$ doesn't have nonnegative determinant.
If the statement meant to say "all its square submatrices (not just those along diagonal) are positive semi-definite", then matrix $[2,-1;-1,2]$ would still be a positive semidefinite matrix but the square submatrix $[-1]$ isn't positive semi-definite.
So I wonder what the statement might actually want to mean? What is the closest correct statement to it?
- Is the converse of the statement also true, i.e. "for a Hermitian or symmetric matrix, if all its square submatrices (not just those along diagonal) are non-negative, then it is positive semi-definite"?
Thanks!