# eigenvalues of qudratic matrix

• Sep 8th 2009, 09:07 AM
eigenvalues of qudratic matrix
Let l
1, l2 . . . , ln be the eigenvalues of a matrix A. What are the eigenvalues of A^2?

there must be some relation bwn. the eigenvalues of A and A^2 matrix which i cannot see it.. Please assist.

Many thanks.

• Sep 8th 2009, 11:10 AM
ThePerfectHacker
Quote:

Let l
1, l2 . . . , ln be the eigenvalues of a matrix A. What are the eigenvalues of A^2?

there must be some relation bwn. the eigenvalues of A and A^2 matrix which i cannot see it.. Please assist.

Many thanks.

If $\displaystyle l$ is eigenvalue of $\displaystyle A$ it means $\displaystyle A\bold{x} = l\bold{x}$ for some eigenvector $\displaystyle \bold{x}$. Therefore, $\displaystyle A^2\bold{x} = A(A\bold{x}) = A(l\bold{x}) = lA\bold{x} = l^2\bold{x}$. Thus, $\displaystyle \bold{x}$ is an eigenvector for $\displaystyle A$ with eigenvalue $\displaystyle l^2$.
• Sep 8th 2009, 11:16 AM
Bruno J.
Suppose $\displaystyle \lambda$ is an eigenvalue associated to some vector $\displaystyle v$; then $\displaystyle Av=\lambda v$, and $\displaystyle A^2v=\lambda Av = \lambda^2 v$, so $\displaystyle \lambda^2$ is an eigenvalue of $\displaystyle A^2$.

Note that the converse does not always hold; $\displaystyle A$ could have no eigenvalues while $\displaystyle A^2$ could. For instance if $\displaystyle A$ rotates the plane by an angle of $\displaystyle \pi/2$, then $\displaystyle A$ has no (real) eigenvalues but $\displaystyle A^2$ does.