A lot of matrix multiplication

Question

So I have a set of 15 $4\times 4$ matrices which I call $X_i, (i=1,2..15)$ and a set of 6 $4\times 4$ matrices which I call $y_j, (j=1,2...6)$. Now I have to calculate $(X_i y_j)-(y_j X_i^*)$ for different values of $i$ and $j$, where $*$ stands for Complex conjugate. Is there a simple command (with perhaps loops) in mathematica using which I can calculate all the 90 results instead of multiplying these matrices individually 90 times?

Answer 1

Define some matrices.

Xs = RandomInteger[{1, 4}, {2, 4, 4}];
ys = RandomInteger[{1, 4}, {3, 4, 4}];

Use Outer applied to the indices to produce your derived matrices:

Flatten[
Outer[#1.#2 - #2.Conjugate@#1 &, Xs, ys]
, 1]

Flatten provides your result matrices in a single level list.

Or to follow your naming convention of variable names with subscripts:

Define the number of Xs and ys:

numXs = 2;
numys = 3;

Flatten[Outer[
  Subscript[X, #1].Subscript[y, #2] - 
    Subscript[y, #2].Conjugate@Subscript[X, #1] &, Range@numXs, 
  Range@numys], 1]