Real Matrix Minors

Definition

Given a square matrix $A \in \R ^{n \times n}$, a real number $m \in \R $ is a minor of $A$ if

\[ m = \det A_{I, J} \]

for some $I \subset \set{1, \dots , n}$ and $J \subset \set{1, \dots , m}$. The number $a$ is a principal minor if $I = J$. It is called the $(i,j)$ minor if it is

\[ \det A_{{I \setminus \set{i}, J \setminus \set{j}}}. \]