Explicit analytical formulas of arbitrary matrices pseudoinverse

In this paper, we have obtained concise explicit analytical formulas for matrix pseudoinversion that are applicable to any matrices, regardless of their dimension and rank. These formulas may be considered as a generalization of the known analytical formulas for complete-rank matrix pseudoinversion to incomplete-rank matrices. The generalization consists in the use of left-side and right-side zero divisors existing for incomplete-rank matrices. In contrast to uniqueness of the known formulas of pseudoinversion for cases with complete rank equal to the number of rows or columns, this paper shows that pseudoinversion of any matrix may be performed by means of two different analytical formulas, one with a left-side zero divisor, and the other with a right-side one. From a computational point of view, the obtained formulas significantly simplify the existing general algorithm of pseudoinversion based on extending the original matrix by its zero divisors, followed by inverting the extended matrix-construction. The simplification is due to the fact that when using the proposed formulas it is not necessary to invert matrix constructions the dimensions of which are larger than those of the original matrix. Besides, each formula uses only one zero divisor (a left-side one or a right-side one), but not both of them as in the existing extended algorithm, so it is not necessary to calculate the other zero divisor. The explicit form of the proposed analytical formulas allows visual analyzing the results of pseudoinversion. This is important, for example, when the desired pseudoinverse matrix is a multiplier in some product and we need to show that the product is zero. When isolating a pseudoinverse matrix from an inverted block-matrix construction (as in the existing algorithm), in the general case, it is sufficiently difficult or even impossible to analyze properties of the desired pseudoinverse matrix without additional transformations. As specific examples of incomplete-rank matrices, the paper shows the advantages of the proposed explicit analytical formulas over the existing general algorithm with extended matrix-construction. The advantages involve both decreasing the computing time and simplifying the form of pseudoinversion results.