15-00-SubmatrixNotation.tex

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\pmtitle{submatrix notation}
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\begin{document}
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
$a_{i_sj_t}$ 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 $A(\alpha, \beta]$.

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