I am struggling with what my professor has written in my notes, because he hasn't proven this anywhere. In part of a longer proof, this comes up:

"Since Y is an orthogonal matrix, so is $Y^{-1}XY$. It follows that $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ is an orthogonal matrix, having determinant 1, so it is a rotation in the plane W"

$$Y^{-1}XY=\begin{pmatrix} 1 & 0 & 0 \\ 0 & a & b \\ 0 & c & d \end{pmatrix}$$

Is it true in general that if Y is orthogonal, $Y^{-1}XY$ is orthogonal?

This is part of a proof of this theorem: If X is a 3x3 orthogonal matrix with det X = 1, then X is given by rotation around some line L through the origin about some angle.


The question in the title is obviously false by taking $X=0$, say. But note that here $X$ is also orthogonal, and the product of any two or more orthogonal matrices is orthogonal (you should try proving this, it's quite simple). Since both $Y$ and $Y^{-1}$ are orthogonal, it should be clear that so is $Y^{-1}XY$.

| cite | improve this answer | |
  • $\begingroup$ I forgot that X was orthogonal. Is there any way I can delete this question? It makes me look dumb haha $\endgroup$ – Mark T Apr 17 '13 at 0:51
  • $\begingroup$ @MarkT I wouldn't know how, but I'm sure I've seen other questions voluntarily retracted. $\endgroup$ – Erick Wong Apr 17 '13 at 0:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.