Analytical aspects of matrix interpolation problems and its applications

TL;DR

This paper develops a theory of two-variable tensor-product polynomials, explores their algebraic properties, and applies them to matrix interpolation problems, establishing isomorphisms and geometric property preservation.

Contribution

It introduces a new algebraic framework for bivariate polynomials related to matrix interpolation and demonstrates their structural and geometric property preservation.

Findings

Proves the existence and poisedness of the matrix interpolation problem.

Provides explicit formulas for polynomial construction in the considered space.

Establishes isomorphism between polynomial space and matrix space.

Abstract

In this paper, the $mn$-dimensional space of tensor-product polynomials of two variables, of degree at most $(m-1)+(n-1)$, is considered. A theory of two-variate polynomials is developed by establishing the algebra and basic algebraic properties with respect to the usual addition, scalar multiplication, and a newly defined algebraic operation in the considered space. Further, the existence of the considered space is established with respect to the matrix interpolation problem (MIP), $P(i,j)=a_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$, corresponds to a given matrix $(a_{ij})_{m \times n}$ in the space of $m \times n$ order real matrices. The poisedness of the MIP is proved and three formulae are presented to construct the respective polynomial in the considered space. After that, using construction formulae, a polynomial map from the space of $m \times n$ order real matrices to the considered space is defined. Some properties of the polynomial map are investigated and some isomorphic structures between the spaces are installed. It is proved that the considered space is isomorphic to the space of $m \times n$ order real matrices with respect to the algebra structure. The polynomials in the considered space with respect to the MIP's for the given matrices also preserve the geometric properties of the matrices such as transpose, symmetry, and skew-symmetry. Some examples are included to demonstrate and verify the results.