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Proving properties of matrices over Z(2)
Abstract
We prove assorted properties of matrices over
${\mathbb{Z}_{2}}$
, and outline the complexity of the concepts required to prove these properties. The goal of this line of research is to establish the proof complexity of matrix algebra. It also presents a different approach to linear algebra: one that is formal, consisting in algebraic manipulations according to the axioms of a ring, rather than the traditional semantic approach via linear transformations.
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We prove that the range of a symmetric matrix over GF (2) always contains its diagonal. This is best possible in several ways, for example GF (2) cannot be replaced by any other field.
We show that the Gaussian elimination algorithm can be proven correct with uniform extended Frege proofs of polynomial size, and hence feasibly. More precisely, we give short uniform extended Frege proofs of the tautologies that express the following: given a matrix A, the Gaussian elimination algorithm reduces A to row-echelon form. We also show that the consequence of this is that a large class of matrix identities can be proven with short uniform extended Frege proofs, and hence feasibly.
This book, first published in 1991, is devoted to the exposition of combinatorial matrix theory. This subject concerns itself with the use of matrix theory and linear algebra in proving results in combinatorics (and vice versa), and with the intrinsic properties of matrices viewed as arrays of numbers rather than algebraic objects in themselves. There are chapters dealing with the many connections between matrices, graphs, digraphs and bipartite graphs. The basic theory of network flows is developed in order to obtain existence theorems for matrices with prescribed combinatorial properties and to obtain various matrix decomposition theorems. Other chapters cover the permanent of a matrix, and Latin squares. The final chapter deals with algebraic characterizations of combinatorial properties and the use of combinatorial arguments in proving classical algebraic theorems, including the Cayley-Hamilton Theorem and the Jordan Canonical Form. The book is sufficiently self-contained for use as a graduate course text, but complete enough for a standard reference work on the basic theory. Thus it will be an essential purchase for combinatorialists, matrix theorists, and those numerical analysts working in numerical linear algebra.
A certain type of transformation of a set of numbers may be represented as the multiplication of a vector by a square matrix. Repetition of the operation is equivalent to multiplying the original vector by a power of the matrix. Properties of powers of matrices are thus of considerable importance, and many general properties have been established for some time.