How to write an orthogonal matrix as multiplication of two orthogonal and symmetric matrix?

Question

I know an orthogonal matrix is multiplication of at most n Householder reflection, which is both orthogonal and symmetric.But how to write an orthogonal matrix as multiplication of two matrix and they are both orthogonal and symmetric?

Answer 1

Every real orthogonal matrix is orthogonally similar to its real Jordan form, which is a direct sum of $I,-I$ and $2\times2$ rotation matrices. Now we are done because $I=(I)(I),\,-I=(-I)(I)$ and $$ \pmatrix{\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta}=\pmatrix{1&0\\ 0&-1}\pmatrix{\cos\theta&-\sin\theta\\ -\sin\theta&-\cos\theta}. $$