Complexity measures of sign matrices

Complexity measures of sign matrices

What is complexity, and how should it be studied mathematically? In the interpretation that we adopt, there are several underlying common themes to complexity theories. The basic ground rules are these: There is a family F of some mathematical objects under consideration. The elements of some subset S ⊆F are deemed simple. Also, there are certain composition rules that allow one to put together...

More about measures and Jacobians of singular random matrices

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

Alternating sign matrices

Symmetric alternating sign matrices

In this note we consider completions of n×n symmetric (0,−1)-matrices to symmetric alternating sign matrices by replacing certain 0s with +1s. In particular, we prove that any n×n symmetric (0,−1)-matrix that can be completed to an alternating sign matrix by replacing some 0s with +1s can be completed to a symmetric alternating sign matrix. Similarly, any n × n symmetric (0,+1)-matrix that can ...

Tight sign-central matrices

A real matrix A is called sign-central if the convex hull of the columns of à contains the zero vector 0 for every matrix à with the same sign pattern as A. A sign-central matrix A is called a minimal sign-central matrix if the deletion of any of the columns of A breaks the signcentrality of A. A sign-central matrix A is called tight sign-central if the Hadamard (entrywise) product of any two c...