I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes the generated matrix has determinant $1$ and sometimes it has determinant $-1$. (There are only two possible values. From the point of view of orthogonal rotation $det=-1$ means that there is also one additional reflection besides the rotation.)

We can change the sign of $det$ of an orthogonal matrix from minus to plus by changing the sign of any one (or, more generally, any odd number of) column of it.

My question is: given that we generate such random matrices repeatedly, will we introduce some bias in their uniform random nature if every time we choose to revert sign of only specific column (say, always 1st or always last)? Or should we have to pick the column randomly in order to keep the matrices represent random uniformly distributed collection?

I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes the generated matrix has determinant $1$ and sometimes it has determinant $-1$. (There are only two possible values. From the point of view of orthogonal rotation $\det=-1$ means that there is also one additional reflection besides the rotation.)

We can change the sign of $\det$ of an orthogonal matrix from minus to plus by changing the sign of any one (or, more generally, any odd number of) column of it.

My question is: given that we generate such random matrices repeatedly, will we introduce some bias in their uniform random nature if every time we choose to revert sign of only specific column (say, always 1st or always last)? Or should we have to pick the column randomly in order to keep the matrices represent random uniformly distributed collection?