SENSITIVITY ANALYSIS OF ALL EIGENVALUES OF A SYMMETRICAL MATRIX

Given A(x) = [a(ij)(x)] a n-by-n symmetric matrix depending (smoothly) on a parameter x, we study the first order sensitivity of all the eigenvalues lambda(m)(x) of A(x), 1 less than or equal to m less than or equal to n. Under some smoothness assumption like the a(ij) be C-1, we prove that the directional derivatives d-->lambda'(m)(x,d)=lim/t-->o(+)[lambda(m)(x+td)-lambda m(x)]/t do exist and give an explicit expression of them in terms of the data of the parametrized matrix. The key idea to circumvent the difficulties inherent to the study of each lambda(m) taken separately, is to consider the functions f(m)(x), 1 less than or equal to m less than or equal to n, defined as the sums of the m largest eigenvalues of A(x). Based on Ky Fan's variational formulation of f(m) and some chain rule from nonsmooth analysis, we derive an explicit formula for the generalized gradient of f(m) and a computationally useful formula for the directional derivative of f(m). Using these formulas and the relation lambda(m) = f(m)-f(m-1), we then derive the directional derivative of lambda(m). Some properties of this directional derivative as well as an illustrative example are presented.