Note on the Heaviside Expansion Formula.

Included among these is the sum of the squares of the characteristic numbers of P1, i.e., the sum of ihe characteristic numbers of N1 = AA*; this is the well-known unitary invariant Eapdpq of Frobenius. p,q When A is normal AA* = A*A or PU.UI*P1 = U*"P.P,U so that I= U*PU = (U*PU)2. Hence P1 = U*P,U or UP1 = POU. Conversely if UP1 = P1U we have A*A = A*A so that a matrix A is normal. when and only when its polar co6rdinates are commutable, that is, when the polar representations A = P1U, A = UP2 coincide. It may be mentioned that the above considerations are valid also in the real domain. In this case the polar representation is simply the algebraic formulation of the fact, well known for n = 3 from the kinematics of homogeneous linear (non-singular) deformations, that any such deformation may be represented as a superposition of a dilatation and a rotation (the norm AA* of A determining the ellipsoid of dilatation belonging to the deformation A). 1 See, for example, Weyl, H.; Gruppentheorie und Quantenmechanik, Leipzig, pp. 19-23, 1928. 2 See Weyl, H., loc. cit. 3 Since writing the above this problem has been solved and will be treated in a forthcoming note in these PROCEEDINGS.