Boolean Inner product Spaces and Boolean Matrices

TL;DR

This paper introduces Boolean inner product spaces and matrices, explores their properties, and provides results on invariant vectors and matrix powers, advancing the mathematical understanding of Boolean linear algebra.

Contribution

It develops the theory of Boolean inner product spaces and matrices, including dimension theorems and invariant vector characterizations, which are novel in Boolean algebra context.

Findings

Dimension theorem for orthonormal bases

Characterization of invariant stochastic Boolean vectors

Results on powers of stochastic and unitary matrices

Abstract

This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.