Idempotency of linear combinations of an idempotent matrix and a \(t\)-potent matrix that commute. (English) Zbl 1077.15022

Authors’ summary: This paper deals with idempotent matrices (i.e., \(A^2 = A\)) and \(t\)-potent matrices (i.e., \(B^t = B)\). When both matrices commute, we derive a list of all complex numbers \(c_1\) and \(c_2\) such that \(c_{1}A + c_{2}B\) is an idempotent matrix. In addition, the real case is also analyzed.
Reviewer’s addendum: As starting point in the paper the previous investigations J. K. Baksalary and O. M. Baksalary [Linear Algebra Appl. 321, No. 1–3, 3–7 (2000; Zbl 0984.15021); ibid. 341, No. 1–3, 129–142 (2002; Zbl 0997.15011) and J. K. Baksalary, O. M. Baksalary and G. P. H. Styan, ibid. 354, No. 1–3, 21–34 (2002; Zbl 1016.15027)] are used.


15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A27 Commutativity of matrices
Full Text: DOI


[1] Baksalary, J.K.; Baksalary, O.M., Idempotency of linear combinations of two idempotent matrices, Linear algebra appl., 321, 3-7, (2000) · Zbl 0984.15021
[2] Baksalary, J.K.; Baksalary, O.M.; Styan, G.P.H., Idempotency of linear combinations of an idempotent matrix and a tripotent matrix, Linear algebra appl., 354, 21-34, (2002) · Zbl 1016.15027
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