Orthogonal Matrices and Similarity Transforms

Question

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results.

All we are given:

• $U$ is $n\times n$ and orthogonal

• $Ax = y$ for some nonsingular $n\times n$ matrix $A$

• $B = U^T AU$

• $Bu = v$

How are $u$ and $v$ related to $x$ and $y$?

What I want to know is if $A = U BU^T$. I feel like there's not enough information given to solve the problem. The first thing that came to mind was a similarity transform, but there's nowhere near enough information about $A$. I'm really lost at this point.

Answer 1

If $U$ is an orthogonal matrix, $U^{-1} = U^T$. So if $B = U^T A U$, then $U B U^T = U U^T A U U^T = A$. On the other hand, $U^T B U = (U^T)^2 A U^2$, which is not the same as $A$ unless $U^2$ happens to commute with $A$.

Answer 2

We can rephrase $Bu = v$ as $$ U^T A U u = v \implies A(Uu) = A(Uv) $$ So, we have $A(Uu) = A(Uv)$ and $Ax = Ay$. So, $Uu$ and $Uv$ satisfy the same equation as $x$ and $y$.

The most concise way we can phrase this is to say that $x - y$ and $U(u-v)$ are both in the nullspace of $A$. We don't know anything outside of this from the information provided.