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

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Homework Help Overview

The problem involves a 2x2 matrix A with specified eigenvectors and eigenvalues, and the task is to find the expression for (A^4)x, where x is a given vector.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the diagonalizability of matrix A based on its distinct eigenvalues and explore the implications of expressing the vector x as a linear combination of the eigenvectors.

Discussion Status

Participants have offered insights on the necessity of expressing the vector x in terms of the eigenvectors and have noted that determining A itself may not be required for the solution. There is recognition of the importance of the relationship between eigenvalues and eigenvectors in the context of the problem.

Contextual Notes

Some participants question whether sufficient information is available to express x as a linear combination of the eigenvectors, indicating potential constraints in the problem setup.

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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!
 
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The eigenspace is two dimensional since there are two distince eigenvalues. Thus, A is diagonalizable. Then...
 
More simply, <2, 0>= <1, -1>+ <1, 1>. Apply A to that. You don't need to determine A itself.
 
You don't need to find A. (2,0)=(1,-1)+(1,1). How would you find A^4((1,-1))?
 
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?
 
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.
 
Thank you very much you guys are awesome.
 

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