A characterization of normal matrices
Alan J. Hoffman, Olga Taussky · Journal of research of the National Bureau of Standards · 1954
A matrix A is called normal if AA *= A * A , where A * is t he transposed and con jugate mat rix of A . It is kno,,·n t hat for a pair of commuting matricef< , A , B , there exists an ordering of the characteristic roots 'I , . .. , ' n of A and fll, ... ,fin of B, s uch that every polynomial p(A,B ) has as characterist ic roots the 11L1mbers p(a t.fl i). This proper ty is, in genera l, weaker than commutativity , but does imply it if B = A*. I t is sho ·n that t his property a lready impli es commutativity of A and A* if it is assumed t.o hold for only one polynomial, provided the latter is suitably chosen . Polynomials of firs t and second degree are examined for t heir suitability .