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#### Article Title

Groups of Matrices That Act Monopotently

#### Keywords

Invertible Matrices, Matrix Groups, Distinct Eigenvalues, Irreducible Groups, Unitary Group, Monopotent Matrices.

#### Abstract

In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form $\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where $z$ is a complex number, and the matrices $G$ form an irreducible subgroup of $\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue. The authors also consider groups $\Fam{G}$ of matrices of the form $\left[\small{\begin{smallmatrix} G & x\\ 0& 1 \end{smallmatrix}}\right]$, where the matrices $G$ comprise a bounded irreducible subgroup of $\GL(\C)$. When \Fam{G} is not finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue.

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