Suppose $A \in \R ^{m \times n}$. Given sets $I \subset \set{1, \dots , n}$ and $J \subset \set{1, \dots , m}$, the submatrix of $A$ at indices $I$ and $J$ is the matrix whose elements are the $I \times J$ elements of $A$ in order.
We denote the submatrix by $A_{I, J}$