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Hessenberg Transformation Beispiel Essay

1. Introduction

Let denote the space of all n-square matrices over the complex field and let be the symmetric group of degree n. For , the permanent function is defined as:
The permanent function therefore resembles the determinant, which in turn is given by: where denotes the sign function.
Although not as prominent as the determinant, the permanent function is still a well-known matrix function, with many applications in combinatorics and graph theory. However, while the determinant can be easily computed, no efficient algorithm for computing the permanent is known. The difficulty for a direct computation of the permanent leads to the idea of trying to compute it by using determinants. This problem dates back to 1913 in a work by Pólya [1], and it has been under intensive investigation since then. While it is clear that the permanent of a matrix: equals the determinant of the related matrix:

Szegö [2] proved that for , there is no way to generalize this procedure. That is, there is no uniform way of changing the signs in the entries of a matrix in order to obtain a matrix B satisfying .

In [3], Gibson proved that if A is an n-square -matrix, and if the permanent of A can be converted to a determinant by affixing ± signs to the elements of the matrix, then A has at most positive entries.

Later, Little [4] proved that an n-square -matrix can be conveniently represented by means of a bipartite graph and reinterpreted the problem of characterizing convertible matrices as the problem of characterizing bipartite graphs whose 1-factors can be counted by using Pfaffians in the manner suggested by Kasteleyn [5].

The computational complexity for determining if a -matrix is convertible was studied by Vazirani and Yannakakis [6] and Robertson, Seymour, and Thomas [7], leading the latter authors to design a polynomial-time algorithm capable of determining whether or not the permanent of the matrix is convertible into a determinant.

In this article, we consider mostly n-square -matrices with the maximum number of positive entries . For these cases, we present a procedure to determine whether or not a given matrix is convertible. Compared with previously available algorithms, this method does not rely on the associated bipartite graph, and is more efficient. This result is presented in Section 4, where we introduce a new concept: the imprint. Before that, in Section 3, we define Hessenberg-type matrices, generalizing the well-known notion of Hessenberg matrices, and present some preliminary results. Our main results are presented in Section 5. We extend Fonseca’s result [8] by presenting an explicit characterization of convertible Hessenberg-type matrices and corresponding subspaces. We conclude that convertible matrices can be reduced to a basic set.

2. Basic Definitions and Preliminary Results

Let us start with a brief summary of definitions and results that are subsequently used throughout the article.

In the present work, we exclusively consider square matrices, so we will drop the qualifier square in what follows. In general, the order n of the considered matrices is arbitrary, except when explicitly stated otherwise.

A matrix is said to be convertible if there exists a -matrix , such that: where ★ denotes the Hadamard product. As already mentioned in the Introduction, a convertible matrix of order n has at most nonzero entries [3].
For a -matrix S of order n, we define the associated coordinate subspace as:
It is clear that if S is convertible then every element of is also convertible, with respect to the same matrix C; i.e., there exists a -matrix such that:

We will also say that is convertible.

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