Matrix regularity analysis

The paper investigates the problem of analyzing the regularity of multidimensional matrices based on the repetition of significant (non-empty) characters in the matrix cells. Such a repetition means that when the matrix is shifted along one or more of its coordinates, some significant characters are preserved. For each shift repeated r times, the regularity number is entered as the product of rs, where s is the number of significant symbols that persist for all r repetitions of the shift. Two numerical characteristics of matrix regularity are introduced: the regularity sum and the regularity coefficient. The regularity sum is defined as the sum of the regularity numbers for all possible matrix shifts and allows you to compare the regularity of matrices of the same form, i.e. the same dimension and the same size with the same arrangement of non-empty characters. The regularity coefficient allows you to compare the regularity of arbitrary matrices and is defined as the percentage of the sum of the regularity of a matrix to the sum of the regularity of the «most regular» matrix (all significant symbols of which are the same) of the same form. Algorithms for calculating the sum and regularity coefficient of a matrix are proposed and implemented in computer programs. As an applied area, the article uses the analysis of the regular structure of the poems of the ancient Chinese «Canon of Poems» (Shih-ching). The poem is represented by a four-dimensional matrix, its coordinates are a stanza, a line in a stanza, a verse in a line, and a hieroglyph in a verse; blank characters equalize the sizes of verses, lines and stanzas. The article presents generalizing results of computer experiments with all 305 poems of Shih-ching.