# submatrix notation

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

Let $A=(a_{ij})$ be an $m\times n$ matrix with elements from some set, usually taken to be a field for ring. Let $k$ and $r$ be positive integers with $1\leq k\leq m$, $1\leq r\leq n$, $\alpha\in Q_{k,m}$ and $\beta\in Q_{r,n}$. We let $\alpha=(i_{1},\ldots,i_{k})$ and $\beta=(j_{1},\ldots,j_{r})$

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 columns are complementary to $\alpha$ and $\beta$, respectively.

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

Title submatrix notation SubmatrixNotation 2013-03-22 16:13:36 2013-03-22 16:13:36 Mathprof (13753) Mathprof (13753) 5 Mathprof (13753) Definition msc 15-00