In the Journal of the London Mathematical Society (July, 1951, Vol. 26, pp. 221-8), proofs are given of a theorem of McCoy's and of various results that follow from it. McCoy's theorem establishes the necessary and sufficient condition that a number of n × n matrices should be reducible to triangular form by the same transformation T-1( )T, where the matrix T is unitary, (We define a triangular matrix to be one in which all the elements below the diagonal are zero.)