Dr. Katz DEq Homework Solutions 88

Dr. Katz DEq Homework Solutions 88 - 88 N.M. Katz: T hen...

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88 N.M. Katz: Then the matrix of f(-a, -b, -c) on N is /P(O) Q(O) P(1) Q(I!. (6.4.0.9) t @(2) ." ." ." . .." '. It's proper values are thus its diagonal terms. If a = b, then all but one of these diagonal terms is non-zero, and hence zero is a proper value of multiplicity one, which implies that the kernel of g(- a, - b, - c) in N is one dimensional. Thus we may assume a +b, and that a< b, since the problem is symmetric in a and b. As a + b, the matrix (6.4.0.9) has precisely two diagonal terms which vanish, namely P(a) and P(v). The subspace N(<b) of N consisting of polynomials of degree <b is stable under (, and f is invertible on
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