I'm interested to characterize and see matrix norms that are unitary invariant.
I'm familiar with the $\|A\|_{\sigma p}$ norm (see Matrix Mathematics, Dennis S. Bernstein page 548) which is defined by $$ \|A\|_{\sigma p} := \begin{cases} \left( \sum_{i=1}^{\min\{n,m\}}\sigma_i^p(A) \right)^{1/p}\quad \quad 1 \leq p < \infty,\\ \sigma_{\max}(A) \quad \quad p =\infty. \end{cases} $$ This is a unitary norm, and in general any unitary invariant norm would only depend on singular values of matrix $A$ and visa versa [*]
What are some other matrix norms that are unitary invariant? Is there a principled way to construct such a norm, i.e., come up with a function of singular values $$f: \mathbb{R}^{\min\{n,m\}} \to \mathbb{R},$$
that would necessary constitute a norm?
i.e., $\|\cdot\| := f(\sigma_1(A), \ldots, \sigma_{\min\{n,m\}}(A))$ will be a norm?
[*] Given a matrix $A$ and $U\Lambda V^\top$, its SVD, use unitary matrices $U$, $V^\top$ to leave out only the singular values $\Lambda$.
It is known that the unitarily invariant norms are precisely the symmetric gauge functions of singular values (cf. Horn and Johnson, Matrix Analysis, 1/e, pp.438-439, theorem 7.4.24). In other words, $\|A\|_m$ is a unitarily invariant matrix norm if and only if $\|A\|_m=\|(\sigma_1(A),\ldots,\sigma_k(A))^\top\|_v$, where $\sigma_1(A),\ldots,\sigma_k(A)$ are the singular values in the SVD of $A$ (so, if $A$ is $p\times q$, there are $\min(p,q)$ of them) and $\|\cdot\|_v$ is a vector norm that satisfies two additional properties to the norm axioms: