As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries.
Specifically, let be the complex vector space of matrices with entries in , endowed with any norm of our liking. As all norms are equivalent by the finite dimensionality of , we may for definiteness pick
(The fact that this norm is not sub-multiplicative is not relevant.) We recall that is the metric space of compact non-empty subsets of with the Hausdorff distance . The purpose of the present post is to prove continuity of the map that sends to its set of eigenvalues . For this it is sufficient to verify continuity of the map that associates with each matrix its characteristic polynomial. Once this is done, continuity of follows.
Now, it is known that the coefficients of the characteristic polynomial
can be expressed as sums of principal minors of , see e.g. of C.D. Meyer’s beautiful book Matrix Analysis and Applied Linear Algebra (SIAM, 2001). Clearly each principal minor of depends continuously on the entries of and therefore on itself. (The latter is most easily seen from (1).) Hence the same is true for the coordinate vector and, at last, for the characteristic polynomial .