Linear Algebra – Is an orthogonal matrix diagonalizable?

Question

My understanding was that every normal matrix is diagonalizable. However I recently read that this applies to unitary but not necessary to orthogonal matrices.

"For every orthogonal operator of euclidean space there is an orthonormal basis in relation to which matrix of that operator is diag(+/-1,…,+/-1,A1,A2,…,Ak) where Ai are matrices of order 2."

Does this mean that orthogonal matrices aren't diagonalizable in $\Bbb R$?

Answer 1

You might want to try diagonalizing $$\pmatrix{ c & -s \\ s & c} $$ where $c$ and $s$ are the sine and cosine of various angles. Trying it for $\pi/4$ should be enlightening.