submatrix notation

Let $n$ and $k$ be integers with $1\le k\le n$. Denote by$Q_{k,n}$ the totality of all sequences of $k$ integers, where the elements ofthe sequence are strictly increasing and choosen from $\{1,\mathrm{\dots },n\}$.

Let $A=(a_{ij})$ be an $m\times n$ matrix with elements from some set, usuallytaken to be a field for ring. Let $k$ and $r$ be positive integers with$1\le k\le m$, $1\le r\le n$, $\alpha \in Q_{k,m}$ and$\beta \in Q_{r,n}$.We let $\alpha =(i_1,\mathrm{\dots },i_k)$ and $\beta =(j_1,\mathrm{\dots },j_r)$

The submatrix $A[\alpha ,\beta ]$ has $(s,t)$ entry equal to$a_{i_s{j}_t}$ and has $k$ rows and $r$ columns.

We denote by $A(\alpha ,\beta )$ the submatrix of $A$ whose rows and columnsare complementary to $\alpha$ and $\beta$, respectively.

We can also define similarly the notations $A[\alpha ,\beta )$ and $A(\alpha ,\beta ]$.