Notation for removal of row / column from matrix

Question

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 $A$?

Answer 1

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.

Answer 2

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.

Answer 3

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 $s=s_1<\dots<s_k,t=t_1<\dots<t_\ell$ one could denote with $A_{st}$ the result of extracting rows $s_1,\dots,s_k$ from $A$ and then columns $t_1,\dots,t_\ell$ from the result, so $A_{ij}$ in the sense of $j$th element of the $i$th row would be shorthand for $A_{(i)(j)}$ (if one associates $1\times 1$ matrices with numbers).