## A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications.(English)Zbl 1235.15007

A square matrix $$A$$ of order $$n$$ over a field $$F$$ is said to be tripotent if $$A^3= A$$. In this note, the author studies some linear algebra properties of the associative algebra generated by two commutative tripotent matrices $$A$$ and $$B$$.
If Jordan canonical forms of $$A$$ and $$B$$ were used from the very beginning (note that $$A$$ and $$B$$ are diagonalizable), proofs would be much simplified.
By the way the author claims that the linear combinations of two commutative tripotent elements and their products can produce $$3^9= 19$$, $$683$$ tripotent elements. This is not accurate. For example, if $$A= B= I_n$$, then there are only three such tripotent matrices.

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 15A24 Matrix equations and identities 15A27 Commutativity of matrices 15B57 Hermitian, skew-Hermitian, and related matrices

### Keywords:

tripotent matrix; disjoint idempotent decomposition
Full Text:

### References:

 [1] Anderson TW, Statistics and Probability: Essays in Honor of C.R. Rao pp 1– [2] DOI: 10.1016/S0024-3795(00)00225-1 · Zbl 0984.15021 [3] DOI: 10.1080/03081080500473028 · Zbl 1112.15009 [4] DOI: 10.1016/j.laa.2004.01.011 · Zbl 1057.15018 [5] DOI: 10.1016/j.laa.2005.02.027 · Zbl 1077.15022 [6] DOI: 10.1016/S0096-3003(02)00249-7 · Zbl 1030.15023 [7] Graybill FA, Introduction to Matrices with Applications in Statistics (1969) [8] Mathai AM, Quadratic Forms in Random Variables: Theory and Applications (1992) [9] DOI: 10.1016/j.amc.2003.10.027 · Zbl 1070.15009 [10] Özdemir H, An. St. Univ. Ovidius Constanta 16 pp 83– (2008) [11] DOI: 10.1016/j.amc.2008.10.017 · Zbl 1167.15019 [12] DOI: 10.1016/j.amc.2007.11.019 · Zbl 1165.15022 [13] DOI: 10.1016/0024-3795(94)90450-2 · Zbl 0803.46070 [14] Spitkovsky IM, Electron. J. Linear Algebra 15 pp 154– (2006) [15] DOI: 10.1080/03081080903257689 · Zbl 1256.15007 [16] Vasilevsky N, Dokl. Akad. Nauk Ukrain. SSR, Ser. A 8 pp 10– (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.