If $A= (QU_2)\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix} (PV_2)^T$, When will diagonal elements of $\Omega_K, \Lambda$ be singular values?

Question

Suppose we show $$A= (QU_2)\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix} (PV_2)^T$$

where $\Omega_k$ and $\Lambda$ are the diagonal matrices from the SVD of two other matrices.

• When will it be the case that the diagonal elements of $$\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix}$$ are singular values of $A$, and that $$(QU_2)\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix} (PV_2)^T$$ is a SVD of $A$?

Answer 1

The definition of SVD says that if you have three matrices, two unitary, $$W,Z : \cases{WW^T=I\\ ZZ^T=I}$$ and one diagonal $\Gamma$, then they will be the singular value decomposition of matrices $W\Gamma Z^T$ and $Z\Gamma W^T$.

So if you have any matrix $A$ equal to that product of matrices with those properties then those matrices are that matrix's SVD.

$$A=W\Gamma Z^T$$