orthogonal matrices

• Mar 30th 2011, 09:23 PM
Jskid
orthogonal matrices
Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix.

I think I need to show that the columns of A are orthonormal to each other, but I don't know how.
• Mar 30th 2011, 09:25 PM
Drexel28
Quote:

Originally Posted by Jskid
Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix.

I think I need to show that the columns of A are orthonormal to each other, but I don't know how.

You often pick the hardest characterization to use. Try picking another one--solution below--use at your own risk

Spoiler:

You sure you want to look so soon?

Spoiler:

Fine, do it

Spoiler:

Isn't it true that $\displaystyle A$ is real orthogonal if and only if $\displaystyle A^{-1}=A^\top$? So that $\displaystyle (AB)^{-1}=B^{-1}A^{-1}=B^\top A^\top=(AB)^\top$?

• Mar 30th 2011, 09:55 PM
FernandoRevilla
A "tiny" alternative:

$\displaystyle M\in\mathbb{R}^{n\times n}$ is orthogonal iff $\displaystyle MM^{t}=I$

So, $\displaystyle (AB)(AB)^t=\ldots =I$