Schatten p Norm of adjoint matrix

Question

Is it true that $\|A^*\|_p=\|A\|_p$?

Is there a simple argument?

$\| \cdot \|_p$ is the Schatten p norm defined with singular values.

Thanks in advance.

Answer 1

The singular values are found by taking the square roots of the eigenvalues of the matrices $A^{*}A$ or $AA^{*}$ which are Hermitian positive semi-definite. These matrices only have real non-negative eigenvalues. So if you take $A^{*} = (U \Sigma V^{*})^{*}$ you end up with $V \Sigma^{*} U^{*}$ but for real values $*$ is just the transpose and the diagonal doesn't change. So $A^{*} = V \Sigma U^{*}$

Since the Schatten p-norm is defined as

$$\| \cdot\|_{p} = \bigg( \sum_{i=1}^{n} \sigma_{i}^{p}(A) \bigg)^{\frac{1}{p}}$$

we have that $\| A\|_{p} = \|A^{*}\|_{p}$