Trying to understand step in proof about minimizing ||Ax|| subject to ||x|| = 1

Question

Let $A$ = $UDV^T$, where $D$ is diagonal, $U$ and $V$ are orthogonal.

We want to find the $x$ that minimizes ||$Ax$|| subject to ||$x$|| = 1.

1. Minimize ||$UDV^Tx$||.

2. ||$UDV^Tx$|| = ||$DV^tx$|| and ||$x$|| = $||V^tx$||.

And so forth ...

It's presented in the proof without proof, so it must be pretty simple. Yet, I'm having a hard time seeing how to show it.

Answer 1

Since $U$ preserves the norm, it is the same to minimize $\|UDV^t x\|$ and to minimize $\|DV^t x\|$. Since $V^t$ preserves the norm, it is the same to minimize over the constraint $\|x\|=1$ and over the constraint $\|V^t x\|=1$.