Minor criteria for Positive or negative definiteness

Question

Let $$A=\begin{pmatrix} 2 & -4 & 0\\ -4 & 8 & 0\\ 0& 0&-3 \end{pmatrix}$$

Clearly $$\Delta _1=2 >0$$

$$\Delta _2=0 $$

$$\Delta _3=0 $$

Now how can I decide whether $A$ is positive definite or negative definite ? or semi positive definite or semi negative definite ?

Answer 1

You would have to look at all principal minors, not just the leading principal minors. For example discarding the first column and the first row you get a negative principal minor of -24. The matrix is indefinite. Alternatively you could calculate the Eigenvalues, see that there is a positive and a negative Eigenvalue and come to the same conclusion.

Answer 2

Straight out of Wikipedia:

"a Hermitian matrix $M'$is positive-semidefinite if and only if all principal minors of $M$ are nonnegative"

Of course in the case where all minors are - stricly - postive your matrix will be positive definite.

Answer 3

If at least one diagonal element is positive and at least one diagonal element is negative, the matrix is indefinite.