A Remark on Powers of Matrices and Idempotency

Abstract

In 1980, Khatri [Linear Algebra Appl. 33(1980) 57-65] had shown that the Hermitian part (The equation is abbreviated) of a square complex matrix A is idempotent, and has the same rank as A iff A is normal and its nonzero eigenvalues are equal to 1. In the same article, the author applied the Singular Value Decomposition to show that if A is a square matrix with rank A=trace A, and A(superscript s)=A(superscript t), for some positive integer s, t, with s≠t, then A is idempotent. On the other result of the same article, Khatri had shown A(A(superscript *)A)(superscript s)=A(AA(superscript *))(superscript s) for some nonzero integer s iff A is normal or AA(superscript *)=A(superscript *)A. In 1999, J.Groβ extended Khatri’s theorems to get the result [Linear Algebra Appl. 289(1999) 135-139], if A is a complex square matrix, then its Hermitian part H(A) is idempotent iff A is unitary equivalent to a matrix (The equation is abbreviated), where T is nonsingular upper triangular with H(T) is idempotent. In this paper, we will apply Schur Theorem to give a new proof of Khatri result ([1], heorem 2), pointing out some mistakes of the proof of ([1], theorem 3(ii)), and giving a complete proof. Also we will give some remarks for the result of J.Groβ.