Differential Equations and Linear Algebra, 4th Edition

7.6 Multiple Eigenvalue Solutions

In Section 7.3 we saw that if the $n\times n$ matrix A has n distinct (real or complex) eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ with respective associated eigenvectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$, then a general solution of the system

$${\displaystyle \frac{d\mathbf{x}}{dt}}=\mathbf{A}\mathbf{x}$$ (1)

is given by

$$\mathbf{x}(t)=c_1{\mathbf{v}}_1{e}^{{\lambda }_{1}t}+c_2{\mathbf{v}}_2{e}^{{\lambda }_{2}t}+\cdots +c_n{\mathbf{v}}_n{e}^{{\lambda }_{n}t}$$ (2)

with arbitrary constants $c_1,c_2,\dots ,c_n$. In this section we discuss the situation when the characteristic equation

$$|\mathbf{A}-\lambda \mathbf{I}|=0$$ (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 $\lambda$, the eigenvector equation

$$|\mathbf{A}-\lambda \mathbf{I}|\mathbf{v}=\mathbf{0}$$ (4)

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