# Semidefiniteness of symmetric matrix checking main minors?

I know that for a given symmetric matrix $\mathbf{A}$ the matrix is positive definite / negative definite $\Longleftrightarrow$ main minors of $\mathbf{A}$ / $-\mathbf{A}$ are positive.

Is it also true that for the case where all main minors are positive or zero, that I can conclude the semi definiteness? I could only find statements for the case in which all minors are positive.

No, one cannot show semidefiniteness in that way. Consider the case where all main minors are zero. Does that mean that $A$ is positive semidefinite? Or is it $-A$?
$$A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$