# eigenvalues of qudratic matrix

• September 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.

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

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