Matrix genetics, part 5: genetic projection operators and direct sums

TL;DR

This paper explores symmetries and algebraic structures in genetic matrices, revealing connections with projection operators and their implications for bioinformatics and theoretical biology.

Contribution

It introduces a novel algebraic framework using projection operators to analyze genetic matrices and their symmetries, linking mathematical structures with biological phenomena.

Findings

Genetic matrices exhibit symmetrical properties related to projection operators.

Connections between genetic code degeneracy and matrix symmetries are established.

Algebraic properties of genetic matrices have implications for bioinformatics and biology.

Abstract

The article is devoted to phenomena of symmetries and algebras in matrix presentations of the genetic code. The Kronecker family of the genetic matrices is investigated, which is based on the alphabetical matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. The matrix P=[C A; U G] in the third Kronecker power is the (8*8)-matrix, which contains 64 triplets. Peculiarities of the degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix of 64 triplets. Phenomena of connections of this mosaic matrix (and many other genetic matrices) with projection operators are revealed. Taking into account an important role of projection operators in quantum mechanics, theory of digital codes, computer science, logic and in many other fields of applied mathematics, we study algebraic properties and biological meanings of these phenomena. Using of notions and formalisms of theory of finite-dimensional vector spaces in bioinformatics and theoretical biology is proposed on the bases of the described results.