# 7.6 Multiple Eigenvalue Solutions

In Section 7.3 we saw that if the  matrix A has n distinct (real or complex) eigenvalues  with respective associated eigenvectors , then a general solution of the system

 (1)

is given by

 (2)

with arbitrary constants . In this section we discuss the situation when the characteristic equation

 (3)

does not have n distinct roots, and thus has at least one repeated root.

An eigenvalue is of multiplicity k if it is a k-fold root of Eq. (3). For each eigenvalue , the eigenvector equation

 (4)

has at least one nonzero solution v, so there is at least one eigenvector associated with . But an eigenvalue of multiplicity ...

Get Differential Equations and Linear Algebra, 4th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.