Thread: eigenvalue/eigenvector

Show that if a matrix A has an eigenvector x corresponding to the eigenvalue h, then x is also an eigenvector of the matix A^2 and state the corresponding eigenvalue.
Write an analogous result for A^n.

I'm afraid this kind of has me stumped. I have managed to do a couple of questions on eigenvalues and eigenvectors, but I can't apply that knowledge to this problem.

Anyone help?!?

If is an eigenvalue of then it means has a non-trivial solution .
But that means .
Thus, is an eigenvector of corresponding with eigenvalue .

Thank you for the help ThePerfectHacker !

I am wondering if someone can help me with another similar problem. Rather than start a new thread I'll ask here. Please have mercy, I am not sure how to format the math correctly. Here goes...

I have a matrix:
1 -1
1 0

I have got the eigenvalues to be: (1 +/- sqrt3) / 2
or e^(i.pi/3) & e^-(i.pi/3)

I must now show what the corresponding eigenvectors are, and I am not sure of the answer.

Originally Posted by zbrn

Thank you for the help ThePerfectHacker !

I am wondering if someone can help me with another similar problem. Rather than start a new thread I'll ask here. Please have mercy, I am not sure how to format the math correctly. Here goes...

I have a matrix:
1 -1
1 0

I have got the eigenvalues to be: (1 +/- sqrt3) / 2
or e^(i.pi/3) & e^-(i.pi/3)

I must now show what the corresponding eigenvectors are, and I am not sure of the answer.

let k be an eigenvalue of the matrix A..
then the corresponding eigenvector are the matrices of the form

and solve for the x's..
note, they may not be unique. in fact, they form a set called the Eigenspace with respect to k.. (note again that as the name implies, the set is a space..)