I'm struggling with this task:
Let $\mathbf{A} \in \mathbb{C}^{3 \times 3}$ be a Hermitian matrix. Its singular value decomposition is given as follows: $$ \mathbf{A}=\left(\begin{array}{ccc} -\frac{\sqrt{2}}{2} i & 0 & -\frac{\sqrt{2}}{2} i \\ \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} \\ 0 & -1 & 0 \end{array}\right)\left(\begin{array}{ccc} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{ccc} -\frac{\sqrt{2}}{2} i & 0 & \frac{\sqrt{2}}{2} i \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \end{array}\right)^H $$ Derive a spectral decomposition for $\mathbf{A}$ from the singular value decomposition and justify your answer.
What I saw:
- The SVD is given, we have $A = U\Sigma V^H$
- This is equiv. to $AV = U\Sigma$
- From there follows: $Av_1 = u_1\sigma_1$, $Av_2 = u_2\sigma_2$, $Av_3 = u_3\sigma_3$
How do i go from there?