4
$\begingroup$

I have a matrix $A=\begin{bmatrix} -5 & -1 & 2\\ 2 & 0 & -2\\ -6 & -1 & 3\end{bmatrix}$, and I need to find an invertible matrix P and a diagonal matrix D such that $D = P^{-1}AP$. I've found the eigenvalues for the matrix and they are $-3, 1, 0$, so I know the D matrix, but I can't seem to figure out the eigenvectors for the P matrix.

$\endgroup$
  • $\begingroup$ What is your problem with calculating the eigenvectors? $\endgroup$ – user127.0.0.1 Feb 19 '14 at 22:25
  • $\begingroup$ @user127.0.0.1 I honestly don't know I'm just not getting the right answer, the eigenvectors form the columns of Matrix P right? I'm just not getting it $\endgroup$ – Johnathon Pomenrantz Feb 19 '14 at 22:33
4
$\begingroup$

The $P$ matrix is just the matrix of eigenvectors of A. You find the eigenvectors by solving the equation $$(\lambda I-A)\mathbf{x}=\mathbf{0}$$ For eigenvalue $\lambda=0$, you get the associated eigenvector $\begin{bmatrix}1\\-3\\1\end{bmatrix}$

For eigenvalue $\lambda=-3,$ you get the associated eigenvector $\begin{bmatrix}1\\0\\1\end{bmatrix}$

Fr eigenvalue $\lambda=1,$ you get the associated eigenvector $\begin{bmatrix}.5\\-1\\1\end{bmatrix}$

Thus your matrix $P=\begin{bmatrix}1&1&.5\\-3&0&-1\\1&1&1\end{bmatrix}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.