For a matrix, is there a generalization for the argument of a complex number like there is for the norm?

Question

Any square complex matrix can be written in a polar form R U, where U = exp(i T) is unitary and R is positive-semidefinite Hermitian, generalizing the complex number concept. Unitary multiplication simply rotates eigenvalues in the complex plane, leaving the spectral norm invariant. The spectral norm can also be bounded by other matrix norms, which makes the general matrix norm useful. Is there any known general matrix argument, such that some function of the eigenvalues of T are some kind of "spectral argument" (much like how the spectral norm is a type of matrix norm)?