Finding eigenvectors of similar matrices

[Alupsaiu]

If v is in Rⁿ and is an eigenvector of matrix A, and P is an invertible matrix, how would you go about finding an eigenvector w of PAP^-1?
I'm thinking you have to use a fact about similarity?

 

[rmehta]

Just use Pv. Then PAP^-1(Pv) = PAv = P \lambda v = \lambda Pv.

 

[Alupsaiu]

Man I hate how painfully simple problems like these are when you see them done haha...thanks a bunch

 

[rmehta]

No problem. To give some intuition for this, you should think of an invertible matrix P as giving a "symmetry" of Rⁿ. Then conjugation by P is the corresponding symmetry of the space of nxn matrices. That's really why people care about similar matrices. If two matrices are similar, then there is some "symmetry" that can transform one matrix into the other.

 

[HallsofIvy]

Another way to think about it: two matrices are similar if and only if they represent the same linear operator, written in different bases (P is the "change of basis" matrix). They necessarily have the same eigenvalues since changing bases would only change vectors, not scalars.