Prove that $\big(E\cdot X^{tr}\big)_I=X_I\cdot E^{tr}$ where $E$ is an elementary matrix.

Question

Definition

Let $x_1,\ldots,x_k$ be vectors in $\Bbb R^n$ with $k\le n$. Let $X$ be the matrix $X:=[x_1,\ldots,x_k]$. If $I=(i_1,\ldots,i_k)$ is a $k$-tuple of integers such that $1\le i_1,<i_2<\cdots<i_k\le n$, we call $I$ an ascending $k$-tuple from the set $\{1,\ldots,n\}$, and we denote by $$ X_I \text{ or } X(i_1,\ldots,x_k) $$ the $k$ by $k$ submatrix of $X$ consisting of rows $i_1,\ldots,i_k$ of $X$.

Now we consider the following row elementary column operations, where $j\neq l$

1. Exchange columns $j$ and $l$;
2. Replace column $j$ by itself plus $c$ times column $l$;
3. multilpy the column $j$ by $\lambda$.

Now given an elementary row operation, with corresponding elementary matrix $E$, then $E\cdot X$ equals the matrix obtained by applying this elementary row operation to $X$. So one can compute the effect of applying the elementary column operation to $X$ by transposing $X$, premultiplying by $E$ and then trasposing back. Thus the matrix obtained by applying an elementary column operation to $X$ is the matrix $$ \big(E\cdot X^{tr}\big)=X\cdot E^{tr} $$ where the symbol $tr$ indicates the transposing operation.

So now I ask to prove that $$ \pmb{\big(E\cdot X^{tr}\big)_I=X_I\cdot E^{tr}} $$ for any ascending $k$-tuple $I$ and for any elementary operation.

So could someone help me, please?

Answer 1

Let $X$ a matrix $m\times n$ and for some $k\le m$ let be $I:=(i_1,...,i_k)$ an ascending $k$-tupla of the set $\{1,...,m\}$. So with this code we indicate by $X_I$ the submatrix of $X$ identified by the row vectors $i_1,...,i_k$. So we observe that any elementary operation on the columns of $X$ implies an analogous elementary operation on the columns of $X_I$ because the components of $j$-th comlumn of $X_I$ are equal to the $i_1,...,i_k$ components of the $j$-th column of $X$. So we conclude that to select a submatrix $X_I$ and then to make on it an elementary comlumn operation is equivalent to make an elementary column operation on $X$ and then to select the submatrix $X_I$. So we conclude that to select a submatrix $X_I$ and then to make on it an elementary comlumn operation is equivalent to make an elementary column operation on $X$ and then to select the submatrix $X_I$.