Methods of finding the eigenvalues of a real matrix.

Abstract

Certain numerical analyses primarily concern themselves with problems normally found in the subjects classified as Linear Algebra and Matrix Theory. One of the problems is the determination of the spectrum (set of eigenvalues) and the eigenspaces for a square matrix. Considering the matrix equation Ax = AX, the problem is to determine those values of A for which the equation has a nonzero solution X. These values of A are called eigenvalues of A. The problem of finding the eigenvalues for A is equivalent to analyzing when the square homogeneous linear system (A -AI)X = 0 has a nonzero solution X. This can occur when the system has infinitely many solutions which is equivalent to this condition: IA -AIl = 0 (called the characteristic equation). Therefore one way to determine the spectrum of the matrix is to find the roots of the characteristic equation. However this method is inherently unstable since very small errors in the coefficients of characteristic equation lead to large deviations of spectrum of the matrix. Therefore, we try to use some strategies to reduce the original matrix to a specialized matrix having the same eigenvalues but with more accuracy, and less work.