What should be the correct result of this product $(Y-XW)^T(Y-XW)$

Question

i usually come across this equation and I am not able to understand it entirely, here X, W and Y are matrices of size N*D, D*1, and N*1 resp. $(Y-XW)^T(Y-XW)$.

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As per me it should be equal to $Y^TY-Y^TXW-W^TX^TY+W^TX^TXW$ but actually it comes out to be $Y^TY-2W^TX^TY+W^TX^TXW$. I simply could not find any explanation for this.

Answer 1

The reason this works is because $W^TX^TY$ is a $1 \times 1$ matrix. It is symmetric, and it appears either (once transposed and once as-is) or (twice as-is).