Test for Definiteness

Given $n\times n$ matrix $A=[a_{ij}]$, the submatrices $A_k$ where $i,j\le k$

• $A$ is Positive Definite Matrix if all determinants $\det (A_k)$ are positive
• $A$ is Negative Definite Matrices if all determinants $\det (A_k)$ alternate sign with $\det (A_1)<0$ , $\det (A_2)>0$, $\det (A_3)<0$, $\det (A_4)>0$, etc