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Is there some common notation for the result of removing the $i$th row, the $j$th column or both of them from a matrix given $A$?

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    Maybe? $A_{-i}$, $A^{-j}$ and $A_{-ij}$ to denote the matrix $A$ with row $i$, column $j$ and both row $i$ and column $j$ removed, respectively. This is maybe similar to game theory-based notation. – 2017-01-05

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yes , usually denoted as $A_{ij}$. This stands for the matrix $A$ after having removed row $i$ and column $j$. This is usefull to express the determinant of $A$ when doing cofactor expansion.

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    Thanks, but how do you distinguish it from $A_{ij}$ denoting the $j$th element in the $i$th row? – 2017-01-05
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    should be clear from the context. For example one can use $a_{ij}$ to denote such element. – 2017-01-05
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How short do you need it? What's also often done is to use selection matrices that select a subset of rows/columns. E.g., to drop the third out of five rows of $A$ you can use $J_{3,5} \cdot A$ where $$J_{3,5} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}.$$

Once you defined what $J_{n,p}$ means in general you can also drop rows and columns via $J_{n,p} \cdot A \cdot J_{m,q}^{\rm T}$ (where $A$ is $p\times q$), apply them sequentially àla $J_{k,p-1} \cdot J_{n,p} \cdot A$ etc.

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    Thanks, that's also nice. I don't need it short, just intended to use it for personal notes and wanted to know how to do it right if there is such thing. – 2017-01-05
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    Okay, good to know. I use such matrices a lot. You can use them for things like structured selections, e.g., $I_n \otimes J_{p,p}$ will drop every $p$-th row from a matrix with $p\cdot n$ rows (here $\otimes$ is the Kronecker product). Stuff like that. – 2017-01-05
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If one defines some notation for the sequence $(1,\dots,i-1,i+1,\dots,n)$, e.g. $\hat i$, it would be also consistent to write $A_{\hat i\hat j}$. The idea is that for sequences $s=s_1<\dots