Diagonalize a matrix using singular value decomposition

Question

Suppose I have a product $A^{T}MA,$ where $A$ and $M$ are $n\times k$ and $% n\times n$ full rank matrices. $A^{T}MA$ is assumed to be diagonal matrix with different entries. I want to find a way to get $A$ that diagnolize $% A^{T}MA.$ For a given matrix $M$ and a preliminary $\hat{A},$ I understand that one way is to use eigenvalues and eigenvectors, i.e., let $\hat{A}^{T}M% \hat{A}=PDP^{-1},$ where $D$ is a diagonal matrix, the by letting $A=\hat{A}% P^{-1},$ then $A^{T}MA=P^{-1}\hat{A}^{T}M\hat{A}P^{-1}=P^{-1}PDP^{-1}P=D.$ However, for the eigenvectors and eigenvaluels, it may introduce complex numbers instead of real number. Another way is to use singular value decomposition $\hat{A}^{T}M\hat{A}=UDV,$ then the question is how to define $A$ based on the above singular value decomposition to satisfy $A^{T}MA$ being a diagonal matrix $D.$ Thanks!