Abstract
We study invertibility of big n× n matrices. There exists a number of algorithms, especially in mathematical statistics and numerical mathematics, requiring to invert step by step large matrices which are closely related to each other. Standard inverting methods require O(n) arithmetical operations therefore using of these algorithms for big values of n becomes problematic. In this paper we introduce some classes of matrices that can be inverted by O(n) operations if we use inverse matrices of other closely related matrices. The most important among them are matrices having big common submatrix and modified sample covariance matrices. We apply our theoretical results constructing a fast algorithm for prediction. This algorithm demonstrates the advantage of our inverting methods and can be used, for example, for safety control in the plant.
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