What is the solution for (A^4)x with given eigenvectors and eigenvalues?

[bluewhistled]

Homework Statement

A is a 2x2 matrix with eigenvectors 1,-1 and 1,1 with respective eigenvalues 2 and 3. x is 2,0. Find (A^4)x

Homework Equations

I know (A^k)v=(lamda^k)v
But I just don't know how to solve this to find A and then multiple it by x

The Attempt at a Solution

See above

Thanks a lot for anyone's help or input!

 

[Vid]

The eigenspace is two dimensional since there are two distince eigenvalues. Thus, A is diagonalizable. Then...

 

[HallsofIvy]

More simply, <2, 0>= <1, -1>+ <1, 1>. Apply A to that. You don't need to determine A itself.

 

[Dick]

You don't need to find A. (2,0)=(1,-1)+(1,1). How would you find A^4((1,-1))?

 

[bluewhistled]

Ah I got it, (A^k)x=(lamda1^4)v1+(lamda2^4)v2

Thanks a lot for pointing out that those two vectors added up to x, I overlooked that. Would this be possible otherwise?

 

[Dick]

If the only information you have about A is it's eigenvectors, and you can't express the vector as a linear combination of eigenvectors, then, no, you don't have enough information about A.

 

[bluewhistled]

Thank you very much you guys are awesome.