# 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 ${\lambda}_{1},\text{}{\lambda}_{2},\text{}\dots ,\text{}{\lambda}_{n}$ with respective associated eigenvectors ${\mathbf{v}}_{1},\text{}{\mathbf{v}}_{2},\text{}\dots ,\text{}{\mathbf{v}}_{n}$, then a general solution of the system

is given by

with arbitrary constants ${c}_{1},\text{}{c}_{2},\text{}\dots ,\text{}{c}_{n}$. In this section we discuss the situation when the characteristic equation

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

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

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