Which common operator performs this transformation?

Question

I am stacking some matrices and I am wondering if there is a well-known name of the following transformation: Given two matrices $A\in\mathbb{R}^{N\times K}$ and $B\in\mathbb{R}^{N\times KM}$, I create matrice $C\in\mathbb{R}^{N\times K(M+1)}$ by expanding B in the sense that I include the $i$-th column of $A$ at the $iM$-th column of $B$.

Example: Let $A = \begin{pmatrix}a_1&a_2 \\a_1&a_2\end{pmatrix}$ and $B = \begin{pmatrix}3&4&5&6\\3&4&5&6\end{pmatrix}$. Then the transformation results in $$C = \begin{pmatrix}a_1&3 &4&a_2&5&6\\ a_1&3&4&a_2&5&6\end{pmatrix}.$$

Can this be represented by some standard matrix multiplication?

Answer 1

Let Scilab help you

-->A = [8 9; 80 90]
 A  =

    8.     9.   
    80.    90.  

-->M = [1 0 0 0 0 0; 0 0 0 1 0 0]
 M  =

    1.    0.    0.    0.    0.    0.  
    0.    0.    0.    1.    0.    0.  

-->A * M
 ans  =

    8.     0.    0.    9.     0.    0.  
    80.    0.    0.    90.    0.    0.  

Now you can find a matrix $N$ such that the result is $C=AM+BN$