# Help with Linear Algebra(Eigenvalues)

• Feb 24th 2008, 11:14 PM
nafix
Help with Linear Algebra(Eigenvalues)
I'm having trouble with a few of the harder linear algebra concepts from the eigenvalue chapter.
1) Using the matrix A= | 3 -2 |
| 1 2 |
verify the Cayley Hamilton theorem.

2) Given the rotation matrix Rtheta = | cos -sin|
| sin cos|
show the matrix has eigenvectors and eigenvalues corresponding to
lambda= e^i*theta : |1|
|-i|
lambda= e^-i*theta |1|
|i |
3)Show that similar matrices A and B have the same eigenvalues. Thus you must show that det(A-lambda*I)= det(B-lambda*I)

Thanks to anyone takes a look at this.
Edit/Delete Message
• Feb 24th 2008, 11:46 PM
Peritus
I'm not saying google should become your best fried but at least make it one of your friends:

Cayleyâ€“Hamilton theorem - Wikipedia, the free encyclopedia

Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia

3. if A and B are similar then there exists and invertible mtrix P such that:

$P^{ - 1} AP = B$

$\begin{gathered}
\left| {B - \lambda I} \right| = \left| {P^{ - 1} AP - \lambda I} \right| = \left| {P^{ - 1} AP - P^{ - 1} \lambda IP} \right| = \left| {P^{ - 1} \left( {A - \lambda I} \right)P} \right| \hfill \\
= \left| {P^{ - 1} } \right|\left| {A - \lambda I} \right|\left| P \right| = \left| {A - \lambda I} \right| \hfill \\
\hfill \\
Q.E.D. \hfill \\
\end{gathered}$