Defect of a Kronecker product of unitary matrices

The generalized defect D(U) of a unitary N x N matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli vertical bar U-i,U-j vertical bar as well as the Gram matrix U*U in the first order. Then the defect d(U) is equal to D(U) - (2N-1), that is the generalized defect diminished by the dimension of the manifold {DrUDc : D-r, D-c unitary diagonal}. Calculation of d(U) involves calculating the dimension of the space in R-N2 spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker product of unitary matrices, thus making it easier to perform calculations numerically. Basing on this, we give a lower bound on D(U) (equivalently d(U)), supposing it is achieved for most unitaries with a fixed Kronecker product structure. Also supermultiplicativity of D(U) with respect to Kronecker subproducts of U is shown. (C) 2011 Elsevier Inc. All rights reserved.