Abstract: This paper introduces a comprehensive theory of matrices whose entries are drawn from the ternary set $$B_{3}$$ = {-1,0,1}. We introduce a base-3 coding scheme that provides a lexicographic ordering of rows and columns, which in turn leads to the definition of canonical ordered matrix classes C3 and D3. Extending classical results for binary regular matrices, we define and characterize ($$k_{+}, k_{-}$$)-balanced matrices as their ternary analogues. We prove that the sequence $$H_{n}$$ = $$3^{n}-1$$ represents the maximum attainable row or column code in this framework. Furthermore, we establish the core structural properties and symmetries of these matrix classes. These results provide a rigorous mathematical foundation for modeling systems with ternary interactions, including signed networks, neural connectivity patterns, and other ternary-structured systems.

Keywords: Ternary matrices, balanced matrices, lexicographic order, signed graphs, matrix coding, combinatorial sequences, Hn numbers

Pages: 20-28

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