# Lower bounds for orthogonal matrix multiplication

Is it possible, according to the current state of knowledge, that orthogonal matrices can be multiplied faster than arbitrary matrices?

More precisely, let $$T(N)$$ denote the worst-case time of the fastest algorithm to multiply two arbitrary $$N\times N$$ matrices (in the real RAM model). Let $$T_{\rm O}(N)$$ denotes the worst-case time to multiply two orthogonal $$N\times N$$ matrices. Is there a proof that $$T_{\rm O}(N) = \Omega(T(N))$$?

It seems intuitive to me that $$T_{\rm O}(N) = \Theta(T(N))$$, but I couldn't come up with any proof. Below are my thoughts and failed attempt.

I became interested in this question at the prompting of a question on math stack exchange. It is easy to show that, for example, $$T_{\rm S}(N) = \Omega(T(N))$$, where $$T_{\rm S}$$ is the worst-case time to multiply two symmetric $$N\times N$$ matrices. This follows directly by reduction by embedding the arbitrary matrices $$A$$ and $$B$$ into the larger symmetric product

$$\begin{bmatrix} 0 & A \\ A^\top & 0 \end{bmatrix}\begin{bmatrix} 0 & B^\top \\ B & 0 \end{bmatrix} = \begin{bmatrix} AB & 0 \\ 0 & B^\top A^\top \end{bmatrix}.$$

However, the analogous emebedding of a matrix $$A$$ (normalized to be a contraction) into an orthogonal matrix

$$A \mapsto \begin{bmatrix} A & (I - AA^\top)^{1/2} \\ (I - A^\top A)^{1/2} & -A^\top \end{bmatrix}$$

requires at least $$\Omega(T(N))$$ operations to construct any way I am aware of, so I was unable to prove the result by the same embedding trick.

Is there a more clever proof that $$T_{\rm O}(N) = \Omega(T(N))$$?