Special cases in the classification of commutative matrix subalgebras of length $n-2$

O. V. Markova<sup>ab

a</sup> Lomonosov Moscow State University
^b Emperor Alexander I St. Petersburg State Transport University

Abstract: Commutative subalgebras of length $n-2$ in the matrix algebra of order $n$ can be subdivided into general algebras, containing a matrix with the maximal possible degree of the minimal polynomial equal to $n-1$, and special cases of algebras for which the degree of the minimal polynomial of any matrix does not exceed $n-2$. In the paper, it is shown that special subalgebras exist only in the algebra of $4\times 4$ matrices over fields of characteristic $2$. A description of such algebras, up to similarity, is obtained. For general algebras in the algebra of $4\times 4$ matrices, a description, up to similarity, over arbitrary fields also is obtained.

Key words and phrases: length of an algebra, commutative matrix subalgebra, nonderogatory matrix, partition of a natural number.

UDC: 512.643

Received: 01.10.2025