# Proof: Eigenvector of B Belonging to \lambda for A*S*x

• WTFsandwich
So, in summary, the homework statement is trying to find an eigenvector of B corresponding to a specific eigenvalue, and the attempt at a solution shows that this can be done by multiplying by the eigenvector corresponding to the given eigenvalue. Next, the student is stuck on how to show that this corresponds to the desired result, but with the help of the previous information, they are able to conclude that A(Sx) = \lambda(Sx) and that this equation corresponds to the desired mathematical statement.f

## 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 $$\lambda$$.

## 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?

## 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 $$\lambda$$.

## 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?
That's a decent start. Next, show that A(Sx) = $\lambda$x. That's what it means to say that Sx is an eigenvector of A corresponding to $\lambda$.

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.

That's a decent start. Next, show that A(Sx) = $\lambda$x. That's what it means to say that Sx is an eigenvector of A corresponding to $\lambda$.
Slight correction: You want to show that A(Sx) = $\lambda$(Sx)
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.
Multiply by x now.

I think I got it now.

After multiplying by x, I have ASx = SBx.

Bx has already been shown equal to $$\lambda$$x, so I substitute that in, giving

ASx = S$$\lambda$$x

$$\lambda$$ can be moved to the other side of S since it's a scalar, giving ASx = $$\lambda$$Sx.

Right.