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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 Matrix inversion, generalized inverses 15A24 Matrix equations and identities 15A27 Commutativity of matrices 15B57 Hermitian, skew-Hermitian, and related matrices
##### Keywords:
tripotent matrix; disjoint idempotent decomposition
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