Numerical methods for finding eigenvalues and eigenvectors of a matrix



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Numerical methods for finding eigenvalues and eigenvectors are separated into two groups, iterative methods and direct methods. The iterative methods, by some algorithm, find an eigenvalue or vector of the matrix} reduce the matrix to a Jacobi (tridiagonal) matrix from which all the eigenvalues and vectors may be found; or reduce the matrix to almost triangular form which may or may not have the eigenvalues as the elements of the principal diagonal. If the matrix does not have the eigenvalues as elements of the principal diagonal, another method is used to find the eigenvalues and vectors of the new matrix and relate them to the original matrix. These methods are subdivided into methods suitable for real symmetric or Hermitian matrices, real nonsymmetric matrices, and arbitrary complex matrices. These methods are generally preferred for matrices of order greater than five. Deflation methods are used with the iterative methods which determine only one eigenvalue or vector at a time. These reduce the nth order matrix to one of order (n-1) whose eigenvalues and vectors are the same or can be related to those of the original matrix, except for the eigenvalue and vector already found. The direct methods determine the coefficients of the characteristic polynomial or evaluate the determinant for approximations of an eigenvalue. These are useful for Jacobi matrices, almost triangular matrices, certain matrices where many of the elements are zero, or small matrices.