## Orthogonal Matrices

Hi Professor Kim,

I looked at the lecture notes. In Lecture 27 there was a proposition talked about On(R), ie Capital letter O sub n over all real numbers. What is that?

Thank you

Mi Sun

$O_n(\mathbb{R})$ is the set of $n\times n$ matrices $A$ with real entries that satisfy the identity

$A^TA=AA^T=I$

That is,

$A^T=A^{-1}$

You can easily check that this is equivalent to the columns of $A$ forming an orthonormal set of vectors. Hence, the terminology *orthogonal matrices*. Also, if you take on $\mathbb{R}^n$ the standard inner product, orthogonal matrices are exactly the ones that are isometries when considered as linear transformations. Incidentally, isometries on general real inner product spaces are also sometimes called orthogonal transformations .

MK