Sylvester criterion for non-symmetric matrices

Question

Is it true that if leading principal minors of a non-symmetric matrix alternate in sign, then real parts of all eigenvalues of the matrix are negative? Sylvester criterion says that it is true for a symmetric case, but I can't find a comfirmation for a general one. Thank you!

Answer 1

As pointed out in a comment, this is not even true for symmetric matrices. Sylvester's criterion only implies that a Hermitian matrix is negative definite when the signs of the leading principal minors alternate in the pattern of $-+-+\cdots$. If the sign pattern is $+-+-\cdots$ instead, the matrix is necessarily indefinite (unless in the scalar case).

Sylvester's criterion does not apply when the matrix is not Hermitian. E.g consider the matrix $\pmatrix{-1&-3\\ 1&2}$. Even the sign pattern for its leading principal minors is $-+$, its two eigenvalues are $\frac{1\pm i\sqrt{3}}2$, which have positive real parts.