On completely normal (0, 1)-matrices and symmetrizability

AbstractIf S1,…, Sm are subsets of a set S, there exists a partition of S into basic sets AI, defined by Boolean operations on the Si, of which the latter are sums. If S is a finite-measure space, and the Si are measurable, the measures of the AI are uniquely determined by those of all the products of the Si. An n × n(0, 1)-matrixA is called “completely normal” (c.n.) if all its 2n corresponding row and column inner products are identical, and “completely normalizable” in case AQ is c.n. for some permutation matrix Q. It follows that A is c.n. iff its rows are identical with its columns in some order, and is completely normalizable iff Aτ = PAQ for permutation matrices P, Q. An algorithm is given for construction of all c.n. matrices, and examples show there are normal matrices not completely normalizable, and c.n. matrices not symmetrizable. Whether every (n, k, λ) matrix is symmetrizable, or even completely normalizable, is not known. It is shown at least that every finite projective plane based on a commutative Veblen-Wedderburn system has a symmetrizable incidence matrix