New formulas for inverting square matrices divided into rectangular blocks by rows or columns are obtained. Along with Frobenius formulas, where diagonal blocks are square, the new formulas constructed using matrix zero divisors make it possible to simplify inverting a block matrix of large dimensions by inverting two matrices of smaller dimensions. The formulas are applicable for inverting matrices written both in numerical (real or complex) and analytical (symbolic) form. For a certain wide class of linear time-invariant dynamic systems, compact analytical algorithms for calculating feedback matrices at solving control problems and evaluating components of the state vector are obtained using new formulas of block inversion. These algorithms are to simplify the generalized formulas of Bass – Gura and Ackermann both in direct and in dual versions. Examples of inverting some matrices divided into rectangular blocks are given for both a numerical matrix with complex-valued elements and a symbolic matrix. The problem of modal control of an aircraft spatial motion is solved using the proposed simplification of the generalized Ackermann formula and zeroing the required components of the controller matrix due to convenient parameterization, which does not affect the poles location.
Зубов et al. (Wed,) studied this question.
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