Question

If X and Y are both matrices, then how does \[(X+Y)^{T}\] evaluate? And if I further extend it to say \[(X+Y)^{T}(X+Y)\] then how would this evaluate too? What are the steps? I don't think it works with the usual algebra rules, right?

Answer 1

(X + Y)^T is X^T + Y^T

and because multiplication is distributive over addition

you can multiply by (X+Y) -> (X+Y)^T X + (A+B)^T Y

Answer 2

So can I say this:
\[(X+Y)^{T}(X+Y) = ((X+Y)^{T})X + ((X+Y)^{T})Y\]
\[=(X^{T}+Y^{T})X + (X^{T}+Y^{T})Y\]
\[=X^{T}X + Y^{T}X + X^{T}Y+Y^{T}Y\]

Do I end here and call this the most simplified?

Answer 3

Yes, I think it's distributable left and right, just so long as your respect the order.