Equality in the Frobenius norm related to the complex Schur decomposition

Question

Let $A \in \mathbb{F}^{n \times n}$, let $X \in \mathbb{F}^{n \times n}$ and let $X=UTU^{*}$ be the complex Schur decomposition, then does the following equality always hold

$$ \| A - UTU^{*} \|^{2}_{F} = \| U^{*} A U - T \|^{2}_{F} $$

with $F$ denoting the Frobenius norm.

Answer 1

Yes. The Frobenius norm is preserved by individual unitary transformations either from the left or from the right as explained in "Approach 1" of this answer to a related question