I'm working with an algorithm that takes as part of its input an orthogonal matrix with real entries. I'm trying to understand how the specification of this matrix affects performance of the algorithm. It's very difficult to do this analytically, so I'm working experimentally for now.
I'm loking for examples of orthogonal matrices that are in some sense 'interesting' or 'extreme'. Obviously the identity matrix is one choice. Another example might be the matrix that is 'least sparse' in the sense that the sum of the absolute values of the entries is as large as possible (is there a name for this?). Another choice is a random sample from the Gaussian orthogonal ensemble.
Can anyone suggest some other examples?
Many thanks.