1. ## Eigenvalue Method

Apply the eigenvalue method to find a general solution to:

x1' = x1 + 2x2 + 2x3
x2' = 2x1 + 7x2 + x3
x3' = 2x1 + x2 + 7x3

Here's what Ive done:

The matrix form of the system is:

x' = 1 2 2
.......2 7 1
.......2 1 7

Then the book says to do this:

A - &I = 1-&....2......2
..............2....7-&....1
..............2.....1.....7-&

I can't follow the rest. Can someone show how to do the rest? Thanks

2. Next thing to do is solve $\displaystyle |A-\lambda I| = 0$

$\displaystyle (1-\lambda)\times [(7-\lambda)(7-\lambda)- 1\times 1] - 2[2(7-\lambda)-1(7-\lambda)] +2[2\times 1-(7-\lambda)\times 2 ]=0$
Then follow the rest of this example

Pauls Online Notes : Differential Equations - Real Eigenvalues

3. Ok I get 3 eigenvalues: 0,6,9

Case 1:
I plug 0 into the matrix from the original post and notcie that its a singular matrix b/c the det is 0. In a 2x2 matrix, this means I can arbitrary choose a and solve for b. How does this work for my 3x3 matrix? Thanks

4. Well, what do you get for your row reduction? Incidentally, you can type up matrices as follows:

$\begin{bmatrix}
1-\lambda &2 &2\\
2 &7-\lambda &1\\
2 &1 &7-\lambda
\end{bmatrix},$
and augmented matrices this way:

$\left[\begin{matrix}1 &0 &0\\ 0 &1 &0\\ 0 &0 &1\end{matrix}\right \left|\begin{matrix}4\\5\\6\end{matrix}\right]$

Just double-click on the matrices to see how I entered them, and then enclose that code in math tags.