Matrix Computations

Abstract

Systems of linear equations are found everywhere in scientific and engineering applications problems; hence, they constitute one of the most important branches of numerical analysis. This chapter is devoted to two algorithms: solution of a linear system by Gauss elimination and computation of its eigenvalues by Leverrier’s method. The Gauss elimination algorithm is implemented first in Section 8.2 in its simplest form. In later sections it’s enhanced to include

• Maximal column pivoting

• Computation of the determinant

• LU decomposition

• Matrix inversion

• Gauss-Jordan elimination to compute the reduced echelon form of a nonsquare or singular matrix

Leverrier’s algorithm first computes the coefficients of the characteristic polynomial and then finds its roots by Newton-Raphson iteration and deflation. You can then use Gauss-Jordan elimination to study the eigenvectors for each eigenvalue. These algorithms and their implementations work with both real and complex scalars.