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\(\{k\}\)-group periodic matrices. (English) Zbl 1116.15004

It is well known that projectors and their generalization have been widely used in different mathematical areas and applications. For a given matrix \(A\in \mathbb{C}^{n\times n}\), a matrix \(X\in\mathbb{C}^{n\times n}\) satisfying \(AXA= A\), \(XAX= X\), and \(AX= XA\) is a group inverse of \(A\). The group inverse of \(A\) is denoted by \(A^{\#}\). Fix \(k\geq 2\). A matrix \(A\in\mathbb{C}^{n\times n}\) satisfying \(A^{\#}= A^{k-1}\) is called a \(\{k\}\)-group periodic matrix. A characterization of such a matrix is: \(A^{\#}= A^{k-1}\) if and only if \(A^{k+1}= A\) for \(k= 2,3,\dots\).
Throughout the paper it is assumed that \(c_1\) and \(c_2\) are nonzero elements of \(\mathbb{C}\) and \(P_1\) and \(P_2\) are nonzero different projectors of the same order over the field \(\mathbb{C}\), that is, \(P_1,P_2\in\mathbb{C}^{n\times n}\setminus\{0\}\) and \(P^2_1= P_1\neq P_2= P^2_2\). Idempotent matrices \(A\in \mathbb{C}^{n\times n}\) are also called (oblique) projectors.
The authors first obtain different characterizations of \(\{k\}\)-group periodic matrices, namely some algebraic characterizations and also some geometrical and topological aspects. Later, they study the problem of finding linear combinations of projectors that are \(\{k\}\)-group periodic matrices, i.e. they describe the set \[ {\mathcal S}(P_1,P_2,k):= \{(c_1, c_2)\in \mathbb{C}^2/(c_1P_1+ c_2 P_2)^{k+1}= c_1 P_1+ c_2P_2\},\;k= 1,2,3,\dots. \] This results extend some well-known results in the literature. The authors introduce a new technique for the general case solved here.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B57 Hermitian, skew-Hermitian, and related matrices
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