A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications

Abstract

A square matrix A of order n is said to be tripotent if A ³ = A. In this note, we give a nine-term disjoint idempotent decomposition for the linear combination of two commutative tripotent matrices and their products. Using the decomposition, we derive some closed-form formulae for the eigenvalues, determinant, rank, trace, power, inverse and group inverse of the linear combinations. In particular, we show that the linear combinations of two commutative tripotent elements and their products can produce 3⁹ = 19,683 tripotent elements.

Keywords:

• idempotent matrix
• involutory matrix
• tripotent matrix
• linear combination
• disjoint idempotent decomposition
• eigenvalues
• determinant
• rank
• trace
• inverse
• group inverse
• quadratic form
• chi-square distribution

AMS Subject Classifications::

• 15A09
• 15A24
• 15A27
• 15B57

Acknowledgements

The author thanks the referee for helpful suggestions and comments.