The Clarke and Michel-Penot subdifferentials of the eigenvalues of a symmetric matrix

We calculate the Clarke and Michel-Penot subdifferentials of the function w hich maps a symmetric matrix to its mth largest eigenvalue. We show these t wo subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized dir ectional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative.