# Eigenvalues/Eigenvectors

• Dec 11th 2007, 04:43 PM
Ideasman
Eigenvalues/Eigenvectors
How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors:

$\displaystyle \lambda_1 = -3$ Eigenvector: [0,1]

$\displaystyle \lambda_2 = 2$ Eigenvector: [1,0]
• Dec 11th 2007, 05:23 PM
Ideasman
Quote:

Originally Posted by Ideasman
How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors:

$\displaystyle \lambda_1 = -3$ Eigenvector: [0,1]

$\displaystyle \lambda_2 = 2$ Eigenvector: [1,0]

Work:

$\displaystyle (\lambda + 3)(\lambda - 2) = \lambda^2 + \lambda - 6$

$\displaystyle (a - \lambda)(d - \lambda) - bc = \lambda^2 + \lambda - 6$

$\displaystyle \lambda^2 - a\lambda - d\lambda + ad - bc = \lambda^2 + \lambda - 6$

I thought the following matrix would work, but it didn't :(:

\displaystyle \left[ \begin {array}{cc} -2&2\\\noalign{\medskip}2&1\end {array} \right]

I got the right eigenvalues, but not the right eigenvectors, grr.
• Dec 11th 2007, 05:35 PM
galactus
See if this helps:

$\displaystyle det\begin{bmatrix}{\lambda}&1\\{\lambda}+6&{\lambd a}+2\end{bmatrix}={\lambda}^{2}+{\lambda}-6$

The correct matrix will work if $\displaystyle Ax={\lambda}x$

Where x is the eigenvector and $\displaystyle \lambda$ is the eigenvalue.

You are given eigenvalues and eigenvectors.
• Dec 11th 2007, 06:13 PM
Ideasman
Quote:

Originally Posted by galactus
See if this helps:

$\displaystyle det\begin{bmatrix}{\lambda}&1\\{\lambda}+6&{\lambd a}+2\end{bmatrix}={\lambda}^{2}+{\lambda}-6$

The correct matrix will work if $\displaystyle Ax={\lambda}x$

Where x is the eigenvector and $\displaystyle \lambda$ is the eigenvalue.

You are given eigenvalues and eigenvectors.

I tried. I can't figure it out.

EDIT: Yay I figured it out. Good old PDP^(-1). The matrix, incase you were wondering, is [[2,0],[0,-3]]