Abstract
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 coecients 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 + P; I + Q, or a polynomial like this I + x1(PQP PQ) + x2(PQPQP PQPQ) + ::: . 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 polynomialsprojectors.
Highlights
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
Summary
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. Ïðè äîêàçàòåëüñòâå òåîðåì 5 è 6 èñïîëüçóåòñÿ ñëåäóþùèé ôàêò, ïîëó÷åííûé â ðàáîòå [3]: åñëè ìíîãî÷ëåí îò äâóõ ïðîåêòîðîâ âèäà (2) äëÿ ëþáîé ïàðû ïðîåêòîðîâ ðàâåí íóëþ, òî âñå êîýôôèöèåíòû ýòîãî ìíîãî÷ëåíà íóëåâûå. ÷òî ýòî óòâåðæäåíèå âåðíî äëÿ ïîðÿäêà ïðîåêòîðîâ m áîëüøå îäíîãî).
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