This paper discusses the polynomials of two projectors that with any selection
of these projectors have the value of the nonsingular matrix.
Results of work [1] about block-triangular form pair of projectors
apply to deduce equations, that the coefficients of always nonsingular
polynomials satisfy to. From the equations is obtained the main result,
namely always nonsigular polynomial can be decomposed into a product of
special polynomials. Special polynomial of two projectors $P,\; Q$
is a linear binomial — $I+\alpha P,\; I+\beta Q$,
or a polynomial like this $I+x_{1} (PQP-PQ)+x_{2} (PQPQP-PQPQ)+\dots$.
It is proved that special polynomials are irreducible.
It turns out that linear binomials can be rearranged with some other special
polynomials. If in a product of special polynomials the linear binomials
are rearranged as much as possible to the left, you will get a product of
special polynomials, called standard. It is proved that the standard form
of product by special polynomials is unigue.
The obtained results have provided a description of the structure of all
polynomials of two projectors that with any selection of these projectors
are nilpotent matrices (nilpotent polynomials). Similar results were obtained
for the involute polynomials and polynomials-projectors.
This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific
works of V. I. Bernik.
Bibliography: 16 titles.
