I'm trying to write down expanding a matrix as an equation, my thought is i have to break it up into odd and even indexes but I'm a little stuck, essentially what I want to write as an equation is:
$$M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$
$$N = \begin{bmatrix} a & a & b & b \\ a & a & b & b \\ c & c & d & d \\ c & c & d & d \\ \end{bmatrix}$$
I want to write Matrix N in terms of matrix M, so i'm thinking I can break it down into:
$$N = \begin{cases} ?? & \mbox{if } i,j\mbox{ is even} \\ ?? & \mbox{if } i,j\mbox{ is odd} \end{cases}$$
but i'm not sure how to proceed from here.
One way to write N in terms of M would be
$N[i,j] = M[\lfloor\frac{i}{2}\rfloor,\lfloor\frac{j}{2}\rfloor]$
But I really don't understand by what you mean by separating it into odd, even indexes.