A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading submatrix inclusively between the $1$x$1$ top-left entry and the matrix itself. For example, $$\begin{bmatrix}
11 & 26 & 41 & 56 & 71 \\
26 & 66 & 106 & 146 & 186 \\
41 & 106 & 171 & 236 & 301 \\
56 & 146 & 236 & 326 & 416 \\
71 & 186 & 301 & 416 & 531
\end{bmatrix}$$ is positive definite if and only if $\begin{vmatrix}
11\end{vmatrix}$, $\begin{vmatrix}
11 & 26 \\
26 & 66
\end{vmatrix}$, $\begin{vmatrix}
11 & 26 & 41 \\
26 & 66 & 106 \\
41 & 106 & 171
\end{vmatrix}$, $\begin{vmatrix}
11 & 26 & 41 & 56 \\
26 & 66 & 106 & 146 \\
41 & 106 & 171 & 236 \\
56 & 146 & 236 & 326 &
\end{vmatrix}$, and $\begin{vmatrix}
11 & 26 & 41 & 56 & 71 \\
26 & 66 & 106 & 146 & 186 \\
41 & 106 & 171 & 236 & 301 \\
56 & 146 & 236 & 326 & 416 \\
71 & 186 & 301 & 416 & 531
\end{vmatrix}$ are each positive. I have been informed that an acceptable alternative is to consider the submatrices between the $1$x$1$ bottom-right entry and the full matrix, i.e., $\begin{vmatrix}
531
\end{vmatrix}$, $\begin{vmatrix}
326 & 416 \\
416 & 531
\end{vmatrix}$, $\begin{vmatrix}
171 & 236 & 301 \\
236 & 326 & 416 \\
301 & 416 & 531
\end{vmatrix}$, $\begin{vmatrix}
66 & 106 & 146 & 186 \\
106 & 171 & 236 & 301 \\
146 & 236 & 326 & 416 \\
186 & 301 & 416 & 531
\end{vmatrix}$, and $\begin{vmatrix}
11 & 26 & 41 & 56 & 71 \\
26 & 66 & 106 & 146 & 186 \\
41 & 106 & 171 & 236 & 301 \\
56 & 146 & 236 & 326 & 416 \\
71 & 186 & 301 & 416 & 531
\end{vmatrix}$. Does mixing these approaches work?
Suppose we label the subdeterminants from the first approach $A_n$, and those from the second approach $B_n$, where $n$ is the number of rows/columns in that subdeterminant. For example, $B_2$ represents $\begin{vmatrix}
326 & 416 \\
416 & 531
\end{vmatrix}$. If I find an arbitrary combination of positive subdeterminants from the $A$ and $B$ approaches, such that all five subscripts are accounted for, i.e., $A_1$, $B_2$, $A_3$, $A_4$, and $B_5$, can I conclude that the original matrix is positive definite?
If so, is it possible to generalize the approach even further? For example, can the subdeterminant $\begin{vmatrix}
171 & 236 \\
236 & 326\end{vmatrix}$ make a contribution, even though it is neither utilized by the $A$ nor the $B$ approach?
Best Answer
To test for positive definiteness, as long as the principal submatrices are nested (and we have submatrices of all sizes, of course), they don't need to be the leading ones or the trailing ones. This is because by relabelling the rows and columns, any nested sequence of principal submatrices can be turned into a sequence of leading principal submatrices.
E.g. for any $S\subseteq\{1,2,3,4,5\}$, let $A(S)$ denotes the principal submatrix $(a_{ij})_{i,j\in S}$. Then $A(\{4\}),\,A(\{2,4\}),\,A(\{1,2,4\}),\,A(\{1,2,4,5\})$ and $A$ form a nested sequence of principal submatrices, and $A$ is positive definite if and only if the determinants of these submatrices are all positive.
In your example, $A_1=A(\{1\},\,B_2=A(\{4,5\}),\,A_3=A(\{1,2,3\}),\,A_4=A(\{1,2,3,4\})$ and $B_5=A$ do not form a nested sequence of principal submatrices. Even if their determinants are all positive, $A$ is not necessarily positive definite. For a counterexample, consider $A=\operatorname{diag}(1,-1,-1,1,1)$.
By the way: