Thread: 2-norm is invariant with respect to orthogonal transformations

Hi,

my book says that it is easy to show that for all orthogonal Q and Z of appropriate dimensions we have

.

I'm thinking that , and .

The 2-norm of a matrix is defined as,



and so

.

I do not know what to do here.. Any tips are greatly appreciated. Thanks.

Where have you used the fact that Q and Z are orthogonal?

Hint: Orthogonal matrices preserve the (usual) inner product (or equivalently the norm).

Here's an attempt..



Since , we get that



That last part looks dodgy...

I don't see clear your third equality. Denote

I'd better use:



Then,



That is,



are similar matrices. As a consequence:



which implies:




Fernando Revilla