Variational Analysis of Spectral Functions Simplified

Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. The argument has a direct analogue for spectral functions of Hermitian matrices, and for singular value functions of rectangular matrices.