Matrix “Feature vectors” and grouping operators in pattern recognition

Problem of grouping information: recovering function, represented by its observations, and the of classification (problem) clusterization problem,  is of great importance for applied research. Choice of math object which represent the object under investigations largely determines the effectiveness: scalars, vectors or objects of other kinds. Such choice is determined by the richness of mathematical structures within which “representatives” are investigated. Euclidean spaces $R^{n}$ are common in this choosing. Euclidean spaces of $R^{m \times n}$ of all $m \times n$ matrices are natural as a math structure for “representatives”, but the handling technique for such spaces is poorer in comparison with vector space. Just the development of the technique handling” for Euclidean space of $R^{m \times n}$, including SVD and Moore-Penrose inversion for the linear operators, constructive construction of orthogonal projectors and grouping operators for matrix spaces is the subject of the article. Important “grouping statements” about minimal ellipsoid, which covers elements of fixed sequence of matrices in $R^{m \times n}$ is represented. This statement generalize correspondent results for real valued vectors. “Grouping statements” is proposed to be the base for constructing correspondence distance in solving clusterization problem.