A Simple Proof of Fiedler's Conjecture Concerning Orthogonal Matrices

We give a simple proof that an n×n orthogonal matrix with n ≥ 2 which cannot be written as a direct sum has at least 4n− 4 nonzero entries. 1. The result. What is the least number of nonzero entries in a real orthogonal matrix of order n? Since the identity matrix In is orthogonal the answer is clearly n. A more interesting question is: what is the least number of nonzero entries in a real orthogonal matrix which, no matter how its rows and columns are permuted, cannot be written as a direct sum of (orthogonal) matrices? Examples of orthogonal matrices of each order n ≥ 2 which cannot be written as a direct sum and which have 4n− 4 nonzero entries are given in [1]. M. Fiedler conjectured that an orthogonal matrix of order n ≥ 2 which cannot be written as a direct sum has at least 4n− 4 nonzero entries. Using a combinatorial property of orthogonal matrices, Fiedler’s conjecture was proven in [1]. A (0, 1)-matrixA of order n is combinatorially orthogonal provided no pair of rows of A has inner product 1 and no pair of columns of A has inner product 1. Clearly, if Q is an orthogonal matrix of order n, then the (0, 1)-matrix obtained from Q by replacing each of its nonzero entries by a 1 is combinatorially orthogonal. A quite lengthy and complex combinatorial argument is used in [1] to show that if A is a combinatorially orthogonal matrix of order n ≥ 2 and A cannot be written as a direct sum, then A has at least 4n − 4 nonzero entries. Clearly this result implies Fiedler’s conjecture. In this note we give a simple matrix theoretic proof of Fiedler’s conjecture. Received by the editors on November 10, 1994, and in revised form on November 7, 1995. This research is supported in part by the National Security Agency under grant MDA904-94-H-2051. Copyright c ©1997 Rocky Mountain Mathematics Consortium