Please can someone help understand which matrix operations might assist to prove that $V = S^{−1}·U^T·A$,
Given that $A = U · S · V$ be the singular value decomposition, where U is an m × r column-orthonormal matrix, S is a diagonal r × r matrix, and V is an r × n row-orthonormal matrix.
I tried some basic operations using transpose, orthogonal properties but couldn't derive the result.
Many thanks!
If $U$ is a column orthonormal matrix, then $U^TU=I$, which is to say column $i$ is orthogonal to column $j$ for $i\ne j$, and so on. So if you multiply your decomposition $A=USV$ on the left by $U^T$ you get $$U^TA=U^T(USV)=(U^TU)SV = I S V = SV.$$ Now multiply that on the left by $S^{-1}$ to get $$S^{-1}U^TA=S^{-1}(SV) = (S^{-1}S )V = IV = V.$$ Which is what you wanted.