On square roots and norms of matrices with symmetry properties

The present work concerns the algebra of semi-magic square matrices. These can be decomposed into matrices of specific rotational symmetry types, where the square of a matrix of pure type always has a particular type. We examine the converse problem of categorising the square roots of such matrices, observing that roots of either type occur, but only one type is generated by the functional calculus for matrices. Some explicit construction methods are given. Moreover, we take an observation by N.J. Higham as a motivation for determining bounds on the operator p-norms of semi-magic square matrices.