A note on linear combinations of commuting tripotent matrices

AbstractThe purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix. In the case of real scalars and real symmetric matrices, this problem admits an interesting statistical interpretation. Namely, it is equivalent to the question of when a linear combination of two quadratic forms in normal variables, each distributed as a difference of two independent χ2-variables, is also distributed as such a difference