direct sum and singular value

Question

Suppose that $A$ is an $n \times n$ complex matrix, and $A$ can be written as direct sum of matrices $\Gamma_i$ for $i=1,\ldots,n$ (i.e.: $A = {\Gamma _1} \oplus {\Gamma _2} \oplus \cdots \oplus {\Gamma _n}$). What relationship (if any) exists between singular values of $A$ and singular values of $\Gamma_i$ for $I=1,\ldots,n?$

Answer 1

If $\Gamma_i=U_i\Sigma_iV_i^T$ are SVD's of $\Gamma_i$ ($i=1,\ldots,n$), then obviously $A=U\Sigma V^T$, where $U=U_1\oplus\cdots\oplus U_n$, $\Sigma=\Sigma_1\oplus\cdots\oplus \Sigma_n$, and $V=V_1\oplus\cdots\oplus V_n$.