How to maximize $\| {\bf U x}\|^2$ when ${\bf U}$ is a upper triangular matrix

Question

When ${\bf U}$ is an $N$-by-$N$ complex-valued upper triangular matrix whose diagonal elements are positive real values, how to obtain an $N$-by-$1$ vector ${\bf x}=[x_1\cdots x_N]^T$ with $|x_n|=1$, for all $n=1,...,N$, that maximizes $\| {\bf U x}\|^2$ ?

Answer 1

Set up a Lagrangian for Lagrange multipliers and take the gradient

$$f(\mathbf{x},\lambda) = \mathbf{x^TU^TUx}-\lambda(\mathbf{x^Tx}-1)$$

$$\nabla_{\mathbf{x}}f = 2\mathbf{x^TU^TU}-2\lambda\mathbf{x^T} =0\implies \mathbf{U^TUx} = \lambda\mathbf{x}$$

In other words, the vectors that extremize the function $\mathbf{x^TU^TUx}$ on the constraint $\mathbf{x^Tx}-1=0$ are the eigenvectors of $\mathbf{U^TU}$.