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

\begin{equation}\label{eq:maxnorm}
\|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 \eqref{eq:maxnorm}.) Hence the same is true for the coordinate vector $[\Pi(A)] \in \CC^{n+1}$ and, at last, for the characteristic polynomial $\Pi(A) \in P_n$.