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


That is,


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 .



Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: