By S. J Hammarling
Hammarling Latent Roots and Latent Vectors (University of Toronto Press 1970) (ISBN 0802017096)
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Additional info for Latent roots and latent vectors
173; also reference 8, Vol. 2, p. 212. 1. Obviously we cannot give the number of calculations required for solving the characteristic equation since this will depend on such factors as the method chosen, the number of iterations needed, whether or not complex or multiple roots are present, and the condition of the polynomial. To find a latent vector of B requires n - 2 multiplications (powers of A). To calculate the latent vector of A from this vector requires n(n- 1 ) multiplications. 2) The calculations required in the modified methods are of the same order as those given above.
This, of course, is not in general true. Suppose that in the Danilevsky process we reach the matrix An, given by Latent Roots and Latent Vectors 44 Here the element we wish to divide by in forming Cr-, is zero. Suppose that arj# 0 for some j < r - 1, then we define a similarity transformation that will interchange a,,-, and a,*. I n order to do this we define a matrix S that will interchange the and jth columns of A,-, when we form A,-, S. This means that S is given by ... o . . o . . o o 1 ...
Hence, we require Now B is a common tridiagonal matrix so that its latent roots are given by t f See Appendix I. 19). We reach the interesting conclusion that the Crank-Nicolson method is stable for any choice of r > 0. It is hoped that this example has demonstrated the importance of latent roots in this field. Latent roots and vectors play an extremely important part in the solution of simultaneous differential equations. Only an elementary introduction to the ease of first-order linear equations with constant coeeeients is given here.
Latent roots and latent vectors by S. J Hammarling