Supnick matrix


A Supnick matrix or Supnick array - named after Fred Supnick of the City College of New York, who introduced the notion in 1957 - is a Monge array which is also a symmetric matrix.

Mathematical definition

A Supnick matrix is a square Monge array that is symmetric around the main diagonal.
An n-by-n matrix is a Supnick matrix if, for all i, j, k, l such that if
then
and also
A logically equivalent definition is given by Rudolf & Woeginger who in 1995 proved that
The sum matrix is defined in terms of a sequence of n real numbers :
and an LL-UR block matrix consists of two symmetrically placed rectangles in the lower-left and upper right corners for which aij = 1, with all the rest of the matrix elements equal to zero.

Properties

Adding two Supnick matrices together will result in a new Supnick matrix.
Multiplying a Supnick matrix by a non-negative real number produces a new Supnick matrix.
If the distance matrix in a traveling salesman problem can be written as a Supnick matrix, that particular instance of the problem admits an easy solution.