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On generalized \(m\)-power matrices and transformations. (English) Zbl 1263.15017

A square matrix \(A\) with complex entries is called a generalized \(m\)-power matrix if there are \(m\) distinct complex numbers \(\lambda_1, \lambda_2,\ldots, \lambda_m\) such that \(\prod_{i=1}^m(A+\lambda_iI)=0\). This definition is introduced in the paper and some characterizations of generalized \(m\)-power matrices in terms of the rank of the factors \(A+\lambda_i I\) are given. Particular emphasis is devoted to \(m\)-idempotent and \(m\)-unit-potent matrices and extensions to generalized \(m\)-power transformations are analyzed.

MSC:

15A24 Matrix equations and identities
15B99 Special matrices
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