A Short Note on Kronecker Square Roots

The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60–64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the characterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.