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Ferreyra, David E.; Lattanzi, Marina; Levis, Fabián E.; Thome, Néstor

Parametrized solutions \(X\) of the system \(AXA = AEA\) and \(A^k EAX = XAEA^k\) for a matrix \(A\) having index \(k\). (English) Zbl 1441.15012

Electron. J. Linear Algebra 35, 503-510 (2019).
The authors consider, for given \(n\times n\) complex matrices \(A\) and \(E\), the matrix equations \[ AXA = AEA \quad\text{and}\quad A^kEAX = XAEA^k\,, \] where \(k\) is the index of \(A\), the solutions of which generalise the G-Drazin inverses of \(A\).
The authors provide a necessary and sufficient condition for these equations to have a solution, and their general solution \(X\) is given in terms of a G-Drazin inverse of \(A\). New representations are thereby obtained for the set of all G-Drazin inverses of \(A\). Some applications are discussed.
Reviewer: Thomas Britz (Sydney)

Cited in 1 Review
Cited in 10 Documents

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
15A10 Applications of generalized inverses

Keywords:

matrix equations; index; Drazin inverses; generalized inverses
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References:

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