The discussion revolves around proving that if \( x \) is an eigenvector of matrix \( B \) corresponding to eigenvalue \( \lambda \), then \( Sx \) is an eigenvector of matrix \( A \) also corresponding to \( \lambda \). The relationship between matrices \( A \) and \( B \) is defined by the equation \( B = S^{-1}AS \).
Some participants have provided guidance on how to proceed with the proof, suggesting specific steps to take, such as multiplying by \( S \) and substituting known relationships. There is an ongoing exploration of the connections between the matrices and their eigenvectors, with some participants expressing uncertainty about how to relate their findings back to the original proof goal.
Participants are working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is a focus on deriving results without directly providing solutions.
That's a decent start. Next, show that A(Sx) = \lambdax. That's what it means to say that Sx is an eigenvector of A corresponding to \lambda.WTFsandwich said:Homework Statement
Let B = S^-1 * A * S and x be an eigenvector of B belonging to an eigenvalue \lambda. Show S*x is an eigenvector of A belonging to \lambda.
Homework Equations
The Attempt at a Solution
The only place I can think of to start, is that B*x = \lambda*x.
However, even starting with that, I can't figure out where to go next.
Could someone point me in the right direction?
Slight correction: You want to show that A(Sx) = \lambda(Sx)Mark44 said:That's a decent start. Next, show that A(Sx) = \lambdax. That's what it means to say that Sx is an eigenvector of A corresponding to \lambda.
Multiply by x now.WTFsandwich said:What do I use to show that?
The only new information I've got that might be helpful is that A = S * B * S^-1
Multiplying on the left by S gives A*S = S*B
After doing that, I'm stuck again. I feel like this is the right track, but I don't know how to relate this back to what I'm trying to prove.