Continuity of eigenvalues and spectral radius for a general matrix

Solution 1:

All of these are continuous, since they are the compositions of continuous functions.

The function from the matrix to any coefficient of the polynomial is itself a polynomial on the entries of the matrix, which is continuous. Thus, the function from a matrix to the vector listing the coefficients of the polynomial is continuous. So, the function from a matrix to its characteristic polynomial is continuous.

The function from a characteristic polynomial to its roots is continuous. So, by the continuity of composition, the function from a matrix to its eigenvalues is continuous.

The function $x \mapsto |x|$ is continuous, as is $(x_1,\dots,x_n) \mapsto \max\{x_1,\dots,x_n\}$. So, the function that yields the largest absolute value of an entry of a vector is continuous. So, by composition, the spectral radius function is continuous.

Solution 2:

Before talking about the continuity of eigenvalues as functions, one must be clear about the term "eigenvalue function". What does it really mean? How is it defined?

If A(t) is an nxn matrix in which each entry is a continuous function of the parameter t, then for each t, A(t) has n eigenvalues (constituting the spectrum of A(t)). The spectrum of A(t) is a multiset and it is uniquely determined for each t (via A(t)). However, how to number (or parameterize) the eigenvalues in the set is tricky and difficult.

It is known that if the domain of t is a disc in the complex plane that contains the origin, then it may be impossible to parameterize the eigenvalues as continuity functions. If t belongs to a real interval or if all the eigenvalues are real, then there exists a selection of n continuous functions that are eigenvalues of A(t) for each t. (See Kato's book Perturbation Theory of Linear Operators.)

Note that eigenvalues are always continuous in the topological sense (i.e. the map from matrices to their spectra is continuous). The roots continuity of polynomials is usually in this sense. The topological continuity and functional continuity of eigenvalues (roots) are related, but not the same.