My professor presented the following in class and I have not been able to construct the proof and wondered if someone could explain this please? This is looking at the reconstruction error for PCA. $W$ is $Fxd$ dimensional meaning that the number of features > number of samples.
Subject to the constraints that $W^TW = I$ so that we cannot arbitrarily minimise the Frobenius norm by altering $W$.
The relationship he asserts is the following: $||X-WW^TX||_F = Tr[(X-WW^TX)^T(X - WW^TX)]$
When I consider the matrix $WW^T$ as $F×F$ dimensional, does this matrix have a $d×d$ submatrix that is the identity where there is padding of zeros in the remaining columns/rows? Does this impact the proof in any way?