The approach to systematically eliminate members from the infinite list can best be illustrated with a simple example. Consider (2.1.11) By the Cayley-Hamilton theorem, the second tensor will satisfy its own A characteristic polynomial, i.e., (2.1.12) where (2.1.13) (2.1.14) (2.1.15) (2.1.16) and (2.1.17) Multiplying the characteristic polynomial equation by gives A (2.1.18) Taking the trace of this last expression yields (2.1.19) and this shows that since and are functions of and then k1, k2 k3 tr(A), tr(A2), tr(A3), (2.1.20) Is a function of only these three invariants as well. Indeed repeated applications of the preceding argument would demonstrate that , can be written in terms of tr(A5), tr(A6), … a linear combination of the first three traces of Therefore, by induction A. (2.1.21)
The approach to systematically eliminate members from the infinite list can best be illustrated with a simple example. Consider (2.1.11) By the Cayley-Hamilton theorem, the second tensor A will satisfy its own characteristic polynomial, i.e., (2.1.12) where (2.1.13) (2.1.14) (2.1.15) (2.1.16) and (2.1.17) Multiplying the characteristic polynomial equation by A gives (2.1.18) Taking the trace of this last expression yields (2.1.19) and this shows that since k1, k2 and k3 are functions of tr(A), tr(A2), and tr(A3), then (2.1.20) Is a function of only these three invariants as well. Indeed repeated applications of the preceding argument would demonstrate that tr(A5), tr(A6), … , can be written in terms of a linear combination of the first three traces of A. Therefore, by induction (2.1.21)
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