Groups of matrices with prescribed spectrum - Cigler, Gregor (visokošolska dela, 2005)
The general form of the problem that we discuss in this work is the following. Let ?$\mathcal{G}$? be a (semi)group of ?$n× n$? matrices over the field ?$\mathbb{F}$? such that each matrix from ?$\mathcal{G}$? is (individually) similar to a matrix with a given property ?$\mathcal{P}$?. Is then the (semi)group ?$\mathcal{G}$? simultaneously similar to a (semi)group of matrices all having the property ?$\mathcal{P}$?, i.e., can we find an invertible matrix ?$S \in {\mathbb{F}}^{n× n}$? such that for all ?$X \in \mathcal{G}$? the matrix ?$SXS^{-1}$? has the property ?$\mathcal{P}$?? When the answer is negative in general, we search for additional assumptions under which the (semi)group ?$\mathcal{G}$? is simultaneously similar to a desired (semi)group. In Chapters 2 and 3 we consider triangularizability of matrix (semi)groups. In this case a matrix has the property ?$\mathcal{P}$?, if it is upper triangular and its spectrum satisfy some additional conditions. If ?$\mathcal{G}$? is a matrix semigroup which is triangularized, diagonal entries on a fixed position form a subsemigroup of the multiplicative group ?${\mathbb{F}} \setminus \{0\}$?. In Chapter 2 we study the triangularizability under the assumption that the union of the spectra of all matrices from ?$\mathcal{G}$? forms a group ?$Γ$?. When ?$Γ = \{1\}$? is the trivial group, the well-known Kolchin''s theorem gives the affirmative answer to our problem: Every semigroup of unipotent matrices is triangularizable. We investigate the case where ?$Γ$? is a finite group and we show that Kolchin''s theorem extends only to the case where ?$Γ = \{1,-1\}$?. We give counterexamples for groups ?$Γ$? not contained in the group ?$\{1,-1\}$?. In the search for further extensions of Kolchin''s theorem in Chapter 3 we introduce ?$p$?-property, which is some kind of independency condition of the eigenvalues. We investigate more closely groups of monomial matrices with this property. The main theorem of this chapter is a generalization of Kolchin''s theorem to the groups with ?$2$?-property. Chapter 4 is dedicated to the permutation-like groups, i.e., the finite groups ?$\mathcal{G} \subset \CC^{n × n}$? such that any matrix ?$X \in \mathcal{G}$? is similar to a permutation matrix. In this case a matrix has the property ?$\mathcal{P}$?, if it is a permutation matrix. Since a matrix similar to a permutation matrix is determined by its spectrum, we could describe this property in terms of the spectrum, but the previous description is more transparent. We deal with the question when a permutation-like group is simultaneously similar to a group of permutation matrices. Various examples in this chapter show that in general a permutation-like group does not have to be simultaneously similar to a group of permutation matrices. In fact, there are counterexamples for every ?$n \ge 6$?. The low-dimensional cases ?$n = 2,3,4,5$? are investigated in detail.
Keywords: matematika; mathematics; matrix theory; teorija matrik; teorija grup; group theory; matrične grupe; spectrum; spekter; similarity; Arbeidsgiverforeningen Spekter; podobnost; triangularization; trikotljivost; permutation matrix; permutacijska matrika