A note on square roots of nonnegative matrices

The max-times algebra is the set R+R+ of nonnegative reals with operations ⊕:(a,b)→max⁡{a,b}⊕:(a,b)→max⁡{a,b} and ⊙:(a,b)→ab⊙:(a,b)→ab. We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.