Determine a singular value partition for the matrix $$ \begin{pmatrix} i \sqrt{6} & -3 i & i \sqrt{3} \\ 0 & 0 & 0 \\ \sqrt{6} & 3 & \sqrt{3} \\ \end{pmatrix}\in M_3(\C). $$\vspace{2em}
This is how I started, but need help with how to continue: To determine a singular value decomposition (SVD) of the given matrix $A$, we start by computing the matrix $A^*A$ and its eigenvalues. Then we can use eigenvalue decomposition to find the matrix $V$, and then calculate the matrix $U$ and the singular values.
Calculate $A^*A$: [ A^*A = \begin{pmatrix} -i \sqrt{6} & 0 & \sqrt{6} \\ 3i & 0 & 3 \\ -i \sqrt{3} & 0 & \sqrt{3} \\ \end{pmatrix} \begin{pmatrix} i \sqrt{6} & -3 i & i \sqrt{3} \\ 0 & 0 & 0 \\ \sqrt{6} & 3 & \sqrt{3} \\ \end{pmatrix} = \begin{pmatrix} 7 & 0 & 3\sqrt{2} \\ 0 & 9 & 0 \\ 3\sqrt{2} & 0 & 9 \\ \end{pmatrix} ]
Calculate the eigenvalues of $A^*A$ by solving the equation $\text{det}(A^*A - \lambda I) = 0$: [ \text{it}\begin{pmatrix} 7-\lambda & 0 & 3\sqrt{2} \\ 0 & 9-\lambda & 0 \\ 3\sqrt{2} & 0 & 9-\lambda \\ \end{pmatrix}