Can the product of two orthonormal matrices be an orthonormal matrix?

Question

Suppose you have a real orthonormal matrix $L$.

Are there any real orthonormal matrices $X$, other than $L'$ and the identity matrix such that $Y=LX$ is also an orthonormal matrix?

Answer 1

The product of any two orthonormal matrices is also orthonormal. Let $L,X$ be two such matrices. Then, $(LX)^{*}(LX)=X^{*}L^{*}LX=I$.

To answer your question, yes: pick $X$ to be any (real) orthonormal matrix not equal to $L^*$ or the identity.

Answer 2

suppse L = $\begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta& \cos\theta\end{pmatrix}$ i.e. a rotation of $\theta$ degrees. L is ortho-normal. Now we rotate it $\phi$ more degrees. And we get a rotation of $\theta + \phi.$ All of those matrices are ortho-normal.