I had a question about the matrix normalisation via SVD. Specifically, I have a $T \times r$ real matrix $A$ and $N \times r$ real matrix B. I would like to obtain the normalized $A^*$ and $B^*$ such that 1). $AB'=A^*B^{*\prime}$; 2). $A^{*\prime}A^*/T$ is a $r$-dimenionsal identity matrix; 3). $B^{*\prime}B^*/N$ is a diagonal matrix.
The paper suggested the following construction of $A^*$ and $B^*$. Define $\Sigma_A := A'A/T$ and $\Sigma_B := B'B/N$ and find the SVD of $\Sigma :=\Sigma_A^{1/2}\Sigma_B \Sigma_A^{1/2}$ such that $\Sigma = USV' $, then construct $A^* := A\Sigma_A^{-1/2}U$ and $B^* := B\Sigma_A^{1/2}U$.
I'm able to show that the first two conditions are satisfied. However, I'm not sure why $B^{*\prime}B^*/N = U'\Sigma U$ is a diagonal matrix.
Any explanation of the reason or other normalization methods are appreciated.
Note that $\Sigma_B$ is a positive semi definite matrix. Then this implies that $\Sigma$ is a positive semidefinite matrix. Thus, it has a SVD in which $U = V$.
From there we get $\frac{1}{N}(B^*)^TB^* = U^T\Sigma U = U^TUSU^TU = S$ since $U$ is unitary.