# Finding the eigenvectors in a 3x3 matrix, given 3 eigenvalues!

• May 6th 2008, 06:35 AM
flawless
Finding the eigenvectors in a 3x3 matrix, given 3 eigenvalues!
A= the 3x3 matrix
[0 1 0]
[1 0 0]
[0 0 3]
I worked out the eigenvalues to be 1, -1, 3. How do i find the 3 eigenvectors, as i need them to find an orthogonal matrix. (because i am asked to orthogonally diagonalize this matrix).. Thanks in advance, help will be much appreciated, i have always been stuck on this, on finding eigen vectors in a 3x3 matrix!
• May 6th 2008, 07:24 AM
Isomorphism
Quote:

Originally Posted by flawless
A= the 3x3 matrix
[0 1 0]
[1 0 0]
[0 0 3]
I worked out the eigenvalues to be 1, -1, 3. How do i find the 3 eigenvectors, as i need them to find an orthogonal matrix. (because i am asked to orthogonally diagonalize this matrix).. Thanks in advance, help will be much appreciated, i have always been stuck on this, on finding eigen vectors in a 3x3 matrix!

Let $\displaystyle x = [x_1,x_2,x_3]^T$ and $\displaystyle \lambda$ denote eigenvalue.Use the definition:
$\displaystyle Ax = \lambda x$

So solve the following system of equations, three times, every time with a different $\displaystyle \lambda$, to get the three eigenvectors.
$\displaystyle x_2 = \lambda x_1$
$\displaystyle x_1 = \lambda x_2$
$\displaystyle 3x_3 = \lambda x_3$