Do an Singular Value Decomposition of
$$ \begin{bmatrix} 0 & \sqrt{2} & 0 & \sqrt{2} \\ \sqrt{2} & 0 & \sqrt{2} & 0 \end{bmatrix}$$
I have tried to find it following the regular algorithm but failed. If someone showed me how do this example, I might get it.
Denote $$ A = \begin{bmatrix} 0 & \sqrt{2} & 0 & \sqrt{2} \\ \sqrt{2} & 0 & \sqrt{2} & 0 \end{bmatrix} $$ Step 1:
Find the spectral decomposition of $AA^*$. That is, find a diagonal matrix $D$ and unitary matrix $U$ for which $$ UDU^* = AA^* $$ To get the diagonal entries of $\Sigma$, take the square root of the entries of $D$, so that we now have $$ U\Sigma\Sigma^*U^* = AA^* $$ In this case, you should have $$ \Sigma = \pmatrix{2&0&0&0\\0&2&0&0} $$ And a valid $U$ would be $$ U = \pmatrix{0&1\\1&0} $$ Step 2:
Find a unitary matrix $V$ such that, with $U$ and $\Sigma$ as above, we have $$ U\Sigma V^* = A $$ Or, equivalently, $$ U\Sigma = A V $$ For the $U$ I've chosen, a valid corresponding $V$ is $$ V = \frac{1}{\sqrt 2}\pmatrix { 1&0&0&-1\\ 0&1&-1&0\\ 1&0&0&1\\ 0&1&1&0 } $$