# Homework Help: Repeated Eigenvalues

1. Nov 23, 2012

### XianForce

1. The problem statement, all variables and given/known data
Solve X' = [ [9, 4, 0], [-6, -1, 0], [6, 4, 3]] * X using eigenvalues.

2. Relevant equations
(A - λI) * K = 0
X = eλt

3. The attempt at a solution
Set up the characteristic equation to find eigenvalues. I found a root of multiplicity 2 of λ=3 and another distinct root λ=5.

When setting up equations to solve for the eigenvectors (setting λ= 3) I found:

6k1 + 4k2 = 0
-6k1 -4k2 = 0
6k1 + 4k2 = 0

So there's only a dependency for k1 and k2. So can't I simply find two linearly independent eigenvectors by substituting different values for k3 such as [2, -3, 1] and [2, -3, 2] and use those as two independent solutions? Or does this mean I still have to walk through the steps of using the Kteλt + Peλt form of the solution to find the second solution for the λ=3 root?

2. Nov 24, 2012

### vela

Staff Emeritus
Re: Repeated Eigen Values

Your first method is the right one.