Deflation by elementary rotations for the solution of algebraic eigenvalue problems

Abstract

For the case, that an eigenvector v of an arbitr~ real matrix A is known, several Authors have proposed methods to elimi~ nate the corresponding eigenvalue from the matrix, either by changing it to zero or bY reducing the order of the matrix by one° The present paper derives from the components v[k] of v a set of elementary (two-dimensional) orthogonal transformations~ which transform A into a matrix A I whose last column has only the diagonal element different from zero. The rules are as follows Start with B o = A, then transform for k = 1,2,..o~n~I Bk_ I into B k = U~ Bk_IUk , where U k is obtained from the unit matrix by replacing the following elements The element [k,k] by cos(t[k]) [k+1,k+1] by cos(t[k]) [k,k+1] by sin(t[k]) [k+1 ,k] by sin(t[k]) , and the rotation angle t[k] is determined by tg(t[k]) = ~ v[i] ~ /v[k+1 ] • Then Bn_ I will be the required matrix A I having zeros in the This process may be continued with the matrix A I until after n(n-1)/2 rotations, A is transformed to triangular form (but between a total of n-1 eigenvectors must be computed). This is certainly a nearly trivial modification of Jacobi's or Greenstadt's method. It has however some decisive advantages: a) If not all eigenvalues are wanted, the process may be stopped earlier. b) The convergence behavior of Greenstadt's method is still not quite known, whereas the method given here is iterative only with respect to the computation of the eigenvectors, but this is a well known procedure.