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Homesubmatrix notation

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# 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 $A(\alpha,\beta]$.

## Mathematics Subject Classification

15-00*no label found*

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