# Help with Linear Algebra(Eigenvalues)

• Feb 24th 2008, 11:12 AM
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.
• Feb 24th 2008, 11:48 AM
galactus
For the first part, the Cayley-Hamilton theorem states that a square matrix satisifies its characteristic equation.

The charpoly is $\displaystyle {\lambda}^{2}-5{\lambda}+8=0$

$\displaystyle A=\begin{bmatrix}3&-2\\1&2\end{bmatrix}$

$\displaystyle A^{2}=\begin{bmatrix}7&-10\\5&2\end{bmatrix}$

So, confirm that $\displaystyle A^{2}-5A+8=0$