# Determinant of a positive semi-definite matrix

If $$M$$ is a Hermitian matrix, then $$M$$ is positive definite if and only if its leading principal minors have positive determinant, i.e the following matrices have positive determinant:

• The upper-left 1-by-1 corner of M
• The upper-left 2-by-2 corner of M
• ...
• M.

I was wondering if in the positive semi-definite case the equivalent condition on all the above matrices would be non-negative instead (determinant $$\geq$$ 0).

For a Hermitian matrix $$A$$ to be positive semi-definite, it is necessary for its leading principal minors to be non-negative, but it is not sufficient as the following example shows:

Consider the matrix $$A = \begin{bmatrix} 0 & 0 \\ 0 & -1 \\ \end{bmatrix}$$ with both leading principle minors $$M_{0} = \det(A) = 0*(-1) - 0*0 = 0,$$

$$M_{1} = \det(A_{2,2}) = \det(\begin{bmatrix} 0 & \square \\ \square & \square \\ \end{bmatrix}) = \det(\begin{bmatrix} 0 \end{bmatrix}) = 0,$$ non-negative, but $$A$$ is not positive semi-definite as it has a negative eigenvalue $$-1$$.

To check if a Hermitian matrix $$A$$ is positive semi-definite one has to test if all principal minors (not only the leading principal minors) are non-negative. (proof)

If we look at the example above, the principal minors are $$M_{0,0} = \det(A) = M_0 = 0,$$

$$M_{1,1} = \det(A_{1,1}) = \det(\begin{bmatrix} \square & \square \\ \square & -1 \\ \end{bmatrix}) = \det(\begin{bmatrix} -1 \end{bmatrix}) = -1,$$

$$M_{2,2} = \det(A_{2,2}) = M_1 = 0.$$

We see that $$M_{1,1}$$ is negative, so the matrix is not positive semi-definite.