2.2 Multiple Eigenvalues and the Jordan Canonical Form

A multiple eigenvalue ? of algebraic multiplicity k can have one or several corresponding eigenvectors. The maximal number of linearly independent eigenvectors k _g is called geometric multiplicity of the eigenvalue, which is less or equal to the algebraic multiplicity:

(2.3)

If the algebraic and geometric multiplicities are equal ( k _g= k), then the eigenvalue is called semi-simple. If there is a single eigenvector corresponding to ? ( k _g=1), then the eigenvalue is called nonderogatory.

First, let us consider a nonderogatory eigenvalue ?. There exist linearly independent vectors u ₀, ..., u _{k ?1} satisfying the equations

(2.4)

The vectors u ₀, ..., u _{k ?1} are called the Jordan chain of length k, where u ₀ is the eigenvector and the vectors u ₁, ..., u _{k ?1} are associated vectors. Equations (2.4) can be written in the matrix form as

(2.5)

where

(2.6)

is an m k matrix, which is real or complex depending on the eigenvalue ?, and

(2.7)

is a k k matrix called the Jordan block.

Let us consider an eigenvalue ? having several linearly independent eigenvectors (the derogatory eigenvalue), i.e., k _g>1. In this case there are integers such that

(2.8)

and linearly independent vectors , satisfying the Jordan chain equations

(2.9)

The numbers are...