# Rob Craigen

## Orthogonal Matrices With Zero Diagonal

Date Friday, March 1, 2019

A couple of years ago Robert Bailey (UMN) asked me whether I knew a way to determine for which $n$ there is a real $ntimes n$ orthogonal matrix with zero diagonal, but nonzero in all other positions. At the time I thought the problem to be too simple to take seriously, but found out it was going to take more than a few moment’s reflection. The question falls properly in a class of problems concerning the degree to which specified properties of a matrix are constrained by rather coarse assumptions about the “shape” of the matrix---usually things like the pattern of zeros or the pattern of signs of the (real) entries of the matrix. At first glance this particular case seems straightforward.

The interest in this case comes from a question in graph theory in which one examines the minimum number of distinct eigenvalues possible among matrices associated in a certain way with graphs, in which progress has been slow and in particular it seemed difficult to close the question for the graph obtained by deleting a perfect matching from the complete bipartite graph $K_(n,n)$.

I discuss our solution to the matrix question and some variants.