Finding the general solution of a system of differential equations

[chris_0101]

Homework Statement

The question is:

Homework Equations

I really don't know what to put here but my method is:
-Find det(A-\lambdaI)
-Find the roots of the determinant - which are the eigenvalues
-Solve for (X -(\lambda)I)

I am stuck at this part

The Attempt at a Solution

So I managed to find the determinant, which is:
\lambda^3 -5\lambda^2 -3\lambda +9

The Eigenvalues are:
\lambda1 = 1
\lambda2 = -3
\lambda3 = -3

Now I'm trying to solve for the eigenvector of eigenvalue 1, however manipulating the matrix does not yield anything that I can use to solve for k1 k2 and k3

Any help with this will be greatly appreciated,

Thanks

 

[HallsofIvy]

An eigenvector \begin{pmatrix}x \\ y \\ z\end{pmatrix} corresponding to eigenvalue 1 must, of course, satisfy
\begin{pmatrix}1 & -1 & 0 \\ 0 & -1 & -2 \\ 0 & 2 & -5\end{pmatrix}\begin{pmatrix}x \\ y \\ z\end{pmatrix}= \begin{pmatrix}x \\ y \\ z\end{pmatrix}

\begin{pmatrix}x - y \\ -y- 2z \\ 2y- 5z \end{pmatrix}= \begin{pmatrix}x \\ y \\ z\end{pmatrix}
Which means we must solve x- y= x, -y- 2z= y, 2y- 5z= z. From the first, equation, subtracting x from both sides, -y= 0 so y= 0. Putting that into both of the other equations, z= 0. Therefore, any eigenvector must be of the form <x, 0, 0>.

 

[chris_0101]

Hey, thanks for the response, I now understand what you have mentioned. Now I have run into another issue, finding the third eigenvector. I managed to find the second eigenvector for eigenvalue2, which is <1,4,4>. Now the third eigenvalue is the same as the second so I would assume that the eigenvector will be the same however this is not the case. Using Wolfram alpha, the third eigenvector is <0,0,0> and I don't know how to get this eigenvector. Any help would be great,
Thanks