7.6 Multiple Eigenvalue Solutions

In Section 7.3 we saw that if the n×n matrix A has n distinct (real or complex) eigenvalues λ1, λ2, , λn with respective associated eigenvectors v1, v2, , vn, then a general solution of the system

dxdt=Ax (1)

is given by

x(t)=c1v1eλ1t+c2v2eλ2t++cnvneλnt (2)

with arbitrary constants c1, c2, , cn. In this section we discuss the situation when the characteristic equation

|AλI|=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 λ, the eigenvector equation

|AλI|v=0 (4)

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

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