Continuity of eigenvalues

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 M_n be the complex vector space of n \times n matrices with entries in \CC, endowed with any norm of our liking. As all norms are equivalent by the finite dimensionality of M_n, we may for definiteness pick

(1)   \begin{equation*} \|A\|_{\infty} := \max_{i,j=1}^n{|a_{ij}|} \end{equation*}

(The fact that this norm is not sub-multiplicative is not relevant.) We recall that \mathcal{F} is the metric space of compact non-empty subsets of \CC^n with the Hausdorff distance d_{\mathrm{H}}. The purpose of the present post is to prove continuity of the map \sigma : M_n \to \mathcal{F} that sends A \in M_n to its set of eigenvalues \sigma(A) \in \mathcal{F}. For this it is sufficient to verify continuity of the map \Pi : M_n \to P_n that associates with each matrix its characteristic polynomial. Once this is done, continuity of \sigma = T \circ \Pi follows.

Now, it is known that the coefficients c_0,\ldots,c_{n-1} of the characteristic polynomial

    \[ \DET{(A - \lambda I)} = c_0 + c_1\lambda + \ldots + c_{n-1}\lambda^{n-1} + \lambda^n \]

can be expressed as sums of principal minors of A, see e.g. \S 7.1 of C.D. Meyer’s beautiful book Matrix Analysis and Applied Linear Algebra (SIAM, 2001). Clearly each principal minor of A depends continuously on the entries a_{ij} of A and therefore on A itself. (The latter is most easily seen from (1).) Hence the same is true for the coordinate vector [\Pi(A)] \in \CC^{n+1} and, at last, for the characteristic polynomial \Pi(A) \in P_n.