# Orthogonal and symmetric matrices

Hello,
I guess this is a basic question.
Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing $X+X^{T}$.

But how can I form a matrix which is both symmetric and orthogonal? That is:
$$M=M^{T}=M^{-1}$$.

marcusl
Gold Member
You have implicitly stated that M is real. In this case I think only the identity matrix matches your requirements.

And also diagonal matrix with 1 or -1 at diagonal. Any more?

$$M=M^{T}$$
$$MM^\dagger=I$$
Here $$M^{T}$$ means "transpose", while $$M^\dagger$$ means "conjugate transpose".