  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.
