Properties of Matrices obtained from Identity

Question

Let $I=\delta_{ij}$ be the identity matrix of order $n$ and $J_{k\ell}=I(k\leftrightarrow \ell)$ i.e. $J_{k\ell}$ formed by interchanging the $\ell$ th and $k$ th columns of the identity matrix.

Is there any simple and tricky way to calculate the following actions (or should I do it manually?)

1. $A\times J_{k\ell}$ for $A\in \mathbf{GL}(n,\Bbb R)$.
2. $ J_{k\ell}\times A$.
3. $ J_{st}\times J_{k\ell}\times A$.

and please tell me any other property if you know.

Answer 1

I assume that $A \times B$ simply denotes the usual product $AB$ of two matrices.

Here's an approach that you might like. Note that the $i,j$ entry of $A$ is given by $e_i^TAe_j$, where $e_i$ denotes the $i$th column of the identity matrix (AKA the $i$th standard basis vector) and $A^T$ denotes the transpose of $A$. Note also that $J_{k\ell}^T = J_{k\ell}$, and that $$ J_{k\ell}e_i = \begin{cases} e_k & i=\ell\\ e_\ell & i = k\\ e_i & \text{otherwise}. \end{cases} $$ With that, we find, for instance, that $$ [J_{k\ell}A]_{ij} = e_i^TJ_{k\ell}Ae_j = (J_{k\ell} e_i)^TAe_j = \begin{cases} e_k^TAe_j & i=\ell\\ e_\ell^TAe_j & i = k\\ e_iAe_j & \text{otherwise} \end{cases} = \begin{cases} A_{kj} & i=\ell\\ A_{\ell j} & i = k\\ A_{ij} & \text{otherwise}. \end{cases} $$ In other words, we find that $J_{k\ell}A$ has the same entries as $A$ except the $\ell$th and $k$th rows have been exchanged. So, the action $A \mapsto J_{k\ell}A$ is the exchange (AKA transposition) of these two rows.

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Note that we can find the product of such "$J$ matrices" (for instance, $J_{st}J_{k\ell}$) by noting that $J_{k\ell}$ is a permutation matrix and that a product of permutation matrices corresponds to a composition of the associated permutations.

Answer 2

I assume that by $\times $ you denote the usual matrix product. Then the result of $A \times J_{kl}$ is the matrix $A$ where the $k$-th and $l$-th column are interchangedor in your notation $A \times J_{kl} = A(k\leftrightarrow l)$. Analogously $ J_{kl} \times A$ is the matrix $A$ where line $k$ and $l$ are interchanged. I will demonstrate the former: Let $B_i$ denote the $i$-th column of $B = A \times J_{kl}$. Then $$B_i = \begin{pmatrix} \sum_{m=1}^n A^{1,m} J_{kl}^{m,i} \\ \vdots \\\sum_{m=1}^n A^{n,m} J_{kl}^{m,i}\end{pmatrix}=\begin{cases}\begin{pmatrix} A^{1,l} & \dots & A^{n,l}\end{pmatrix}^T& \text {for } i = k. \\ \begin{pmatrix} A^{1,k} & \dots & A^{n,k}\end{pmatrix}^T& \text {for } i = l. \\ \begin{pmatrix} A^{1,i} & \dots & A^{n,i}\end{pmatrix}^T& \text {else}.\end{cases}$$