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