Solve Repeated Eigenvalues: X' = [[9,4,0], [-6,-1,0], [6,4,3]] * X

[XianForce]

Homework Statement

Solve X' = [ [9, 4, 0], [-6, -1, 0], [6, 4, 3]] * X using eigenvalues.

Homework Equations

(A - λI) * K = 0
X = e^λt

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 Kte^λt + Pe^λt form of the solution to find the second solution for the λ=3 root?

 

[vela]

Your first method is the right one.