## 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.

### MSC:

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

### Keywords:

Idempotent matrix; $$t$$-potent matrix; Linear combination

### Citations:

Zbl 0984.15021; Zbl 0997.15011; Zbl 1016.15027
Full Text:

### References:

 [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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