Строго косые проекторы и их свойства

Abstract

In this paper, strictly oblique projectors are defined as projectors that cannot be represented as the sum of two projectors, one of which is a nonzero orthoprojector. A theorem is proved that each projector can be represented in a unique way as the sum of a strictly oblique projector and an orthoprojector. The properties of such projectors are given. For example: if the projector is strictly oblique, then its Hermitian adjoint is also strictly oblique; the rank of a strictly oblique projector is at most n/2, where n is the order of the projector matrix; the property of the projector to be strictly oblique is preserved with a unitary similarity. The work is a continuation of the previous work of the authors, the main result of which is such a matrix expression for an arbitrary projector: where A and B are two matrices of full rank whose columns define range and the null space of this projector. Based on this result, the article shows that the strictly oblique part of any projector P is given by the expression: P(P – P+P)+P. And equality P = P(P – P+P)+P is a criterion that the projector P is a strictly oblique projector. The decomposition of the projector obtained in the work is applied to the practical problem of oblique projection onto the plane