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simple form for the nilpotent part of . Finding a basis of generalized eigenvectors that reduces
to this form is generally difficult by hand, but computer algebra systems like Mathematica
have built in commands that perform the computation. Finding the Jordan form is not necessary
for the solution of linear systems and is not described by Meiss in chapter 2. However, it is the
starting point of some treatments of center manifolds and normal forms, which systematically
simplify and classify systems of nonlinear ODEs. This subsection follows the first part of
section 1.8 in Perko closely. The following theorem is described by Perko and proved in Hirsch
and Smale:
Theorem (The Jordan Canonical Form). Let
and complex eigenvalues
exists a basis be a real matrix with real eigenvalues and
for eigenvectors of , the first
. The matrix . Then there
, where , of these are real and are generalized
for is invertible and ,
where the elementary Jordan blocks are either of the form , for one of the real eigenvalues of , [2] or of the form [3] 9
Chapter 2 part B with
,
for and , one of the complex eigenvalues of . The Jordan form yields some explicit information about the form of the solution on the initial
value problem
[4]
which, according to the Fundamental Solut...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans

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