SIGNIFICATIONS, DESIGNATIONS OF COMPOSITE MATRIXES THAT IX ANALYSIS

Abstract

The existing algebraic theory of matrices is created for the abbreviated recording of solutions and solutions to problems of linear algebra, that is, it "serves" the performance of operations with linear forms. Because of this, in our study we will call these traditional matrices - "algebraic", in contrast to composite matrices (compomatrices), which are introduced and developed by us.

By numerical field we mean any set of numbers within which are always defined and can unambiguously perform four operations: addition, subtraction, multiplication and division.

Since algebraic matrices are formed over the field K, to determine their elements over the field K must first formulate a problem, determine the initial conditions for it and make the appropriate linear algebraic equations, which determine the elements of algebraic matrices. It follows that the elements of algebraic matrices can’t be arbitrarily selected from the set of the field K. All of them are determined by certain algorithms, as a result, the elements of the algebraic matrix are always combination values, ie such that changing the value of any one of them, entails a change in the values of all its other elements. In other words, the linear transformation problem uniquely defines an algebraic matrix. Conversely, any algebraic matrix uniquely defines a linear transformation.

Composite matrices (compomatrices) are completely different in nature and purpose.

If algebraic matrices are designed for abbreviated writing and compact solution of problems presented by linear algebra in matrix form, computer matrices are designed for analytical formalization of geometric figures by compositional geometry and abbreviated writing and compact solving of geometric problems in analytical form. However, compositional geometry is radically different from analytical geometry in that the equations of geometric objects are formed relative to the basis points of the original GF, for which the problem is solved. Conversely, in analytical geometry, the equations are formed relative to the coordinate system in which the original GF is located.

Keywords: basis points, geometric figure, compositional geometry, compomatrices, compositional geometric modeling, point polynomial.