Matrix Polynomials, Spectral Radius

Spectral Radius

The spectral radius of a matrix is the radius of the smallest circle in the complex plane that contains all its eigen values. Every characteristic polynomial has at least one root, hence every matrix has at least one eigen value. At the same time, a matrix has at most n eigen values, hence the spectral radius is well defined.

Squaring M squares its eigen values. Hence the radius of M2 is the square of its radius, and so on for higher powers.

Majorize

The real matrix B majorizes the complex matrix A if entries of B exceed the absolute values of the corresponding entries of A. If B majorizes A, and D majorizes C, verify that BD majorizes AC. By induction, Bn majorizes An. If Bn approaches 0 than An also approaches 0.

In earlier sections we have seen the utility of switching to a similar matrix. If P is invertible then the limit of PMi/P, as a sequence or series, is Pl/P, where l is the original limit. If a sequence approaches 0, all similar sequences approach 0. We can therefore put M in jordan form, to see if Mn approaches 0. The diagonal elements of Mn are eigen values raised to the nth power, so the spectral radius must be less than 1. Conversely, assume the spectral radius is less than 1, and use the binomial theorem to see that the subdiagonal elements of a simple jordan block become n times the eigen value raised to the n-1. The next diagonal is (n:2) times the eigen value raised to the n-2, and so on. A power of something less than 1, times a polynomial, approaches 0. Everything below the main diagonal approaches 0. Therefore Mn approaches 0 iff the spectral radius of M is less than 1.

Geometric Series

Consider the sum of Mn, as n runs from 0 to infinity. This is a geometric series applied to matrices. If it converges, the terms must approach 0, hence the spectral radius of M is less than 1. Conversely, assume the spectral radius is less than 1. The sum of the first n terms, i.e. the nth partial sum, equals (1-Mn)/(1-M). With Mn approaching 0, the series approaches 1/(1-M).

If M has spectral radius less than 1, 1-M is invertible.