A matrix symmetry property

At this point it is useful to recognize a group property of some 2-by-2 matrices. If F(a,b) represents any 2-by-2 matrix, a function of a and b, with the property that

F₁₁(a,b) = F₂₂(b,a)

and

F₁₂(a,b) = F₂₁(b,a)

then products of two such matrices will also have the property. That is, matrices with this property form a group under matrix multiplication. If we substitute /rho for a and s for b, then our basic building block operator (Equation 8) has this property, and so do the terms in the Taylor series expansion of this operator, and so do any linear combinations of products of these terms. It follows that the terms in the Taylor series expansion of our product forms will also have the property, and this will greatly simplify the development.