Thread: 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

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


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

Pauls Online Notes : Differential Equations - Real Eigenvalues

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

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

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

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

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