SVD decomposition of diagonal matrix $A = \begin{pmatrix}0 & C^T \\ C & 0 \end{pmatrix}$

Question

Assume I have a matrix

$A = \begin{pmatrix}0 & C^T \\ C & 0 \end{pmatrix}$

I want to do sigular value decomposition of $A$. Can I take the benefit of the diagonal property if $A$? like only SVD of $C$ ?

Thanks.

Answer 1

The answer is yes. Suppose that we have the SVD $$ C = U_1\Sigma_1 V_1^T $$ where $U_1,V_1$ are orthogonal matrices. It follows that $A = U_2 \Sigma_2 V_2^T$, where $$ U_2 = \pmatrix{0&V_1 \\U_1 & 0}, \quad \Sigma_2 = \pmatrix{\Sigma_1 & 0\\ 0 & \Sigma_1^T}, \quad V_2 = \pmatrix{V_1 & 0\\0 & U_1}. $$ Or, you could also take $$ U_2 = \pmatrix{V_1&0 \\0&U_1}, \quad \Sigma_2 = \pmatrix{\Sigma_1^T & 0\\ 0 & \Sigma_1}, \quad V_2 = \pmatrix{0 & V_1\\U_1 & 0}. $$ Note that we have $\Sigma_1 \neq \Sigma_1^T$ if $C$ is not square.