Nonnegative Roots of Matrices
Peng-Ruei Huang
Graduate School of Science and Technology
Hirosaki University
h16ds202@hirosaki-u.ac.jp
The root of matrices is a classical problem in matrix theory which can be traced back to the work of Arthur Cayley in 1858. However, not much is known about the question of existence of entrywise nonnegative square roots for a nonnegative matrix. We will consider the nonnegative roots of rank-one matrices and circulant matrices, etc. The necessary and sufficient conditions for the existence of the nonnegative `p^{\text{th}}` root of a circulant matrix with the order 3 and 4 will be given. Moreover, it is proved that the roots of a circulant matrix are circulant if and only if its eigenvalues are all distinct.Keyword: Circulant matrix, nonnegative matrix, matrix roots
References
[1] A. Cayley, A memoir on the theory of matrices, Phil. Trans. Roy. Soc. London, 148 (1858), 17-37.