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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
*F*_{11}(*a*,*b*) = *F*_{22}(*b*,*a*)

and
*F*_{12}(*a*,*b*) = *F*_{21}(*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.

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Stanford Exploration Project

11/17/1997