# Creating an Gaussian Orthogonal Matrix

I am working on a project where I need to create a special type of orthogonal matrix, such that all of the rows and all columns are orthogonal, but each entry in the matrix is drawn from a Gaussian distribution with a mean of $$0$$ and variance of $$1$$.

Is this possible, and if so what would be an easy way to do it?

Here is a partial answer. If your matrix is $$2^n\times2^n$$, we can obtain a random sample from the $$n$$-fold Kronecker product $$\pmatrix{1&-1\\ 1&1}\otimes\cdots\otimes\pmatrix{1&-1\\ 1&1}\otimes\pmatrix{U&-V\\ V&U}$$ where $$U$$ and $$V$$ are two i.i.d. standard normal random variables.