Some Researches On Matrix Mth Roots

In this paper, we discuss some theories of matrix mm roots. Firstly, we talk of the matrix square roots of a general Jordan normal form matrix of J = Jm1(λl) ...Jm1(λl). We obtain the sufficient andnecessary condition for J to have matrix square roots. Furthermore, we obtain a method for computing the matrix square roots by decreasing the order of the Jordan normal form matrix of J = Jm1(λ1)....Jm1(λ1). It is operable. Secondly, we talk of the matrixmth roots of a general Jordan normal form matrix of J = Jm1 (λ1)... Jm1 (λ1) . We obtain the sufficient and necessarycondition for J to have matrix mth roots and a necessary condition for a matrix to be a matrix mth root of J. Lastly, letJ = Jm1 (λ1 ).... Jnk (λk) with λ1...., λk not being zero and not equalingto each other, A(0) =(λ(1)0,....,λ(k)0) is an arbitrary given vector, where λ(i)0(i = 1,....,k) is a given mth root of λi it shows that the matrix mth root of J with its spectrum being {λ(1)0,.....,λ(k) 0} is unique; let J = Jn1(λ) Jn2(λ), λ ≠ 0, x0 is a given mth root of λ,it proves that the matrix mth root of J with spectrum being { x0 } is unique;Furthermore, it gives out a class of nonsingular matrices has unique matrix mth root under given conditions.