ABSTRACT
This chapter reviews elementary matrix operations, find eigenvectors, and invert matrices for solving a system of linear equations. The system of equations appears in finite element analyses and in inverse problems for parameter determination. For stability of matrix inversion, small positive damping factors have frequently been added to the diagonal entries of the to-be-inverted matrix. An ill-conditioned matrix may be encountered in least-squares inverse problems. The technique of singular value decomposition can help us to diagnose singularity and frequently, but not always, achieve a useful numerical solution. If the secular equation has a repeated root, the repeated eigenvalue is degenerative. The eigenvectors associated with a degenerative eigenvalue can be made orthogonal by the Schmidt orthogonalization procedure. Direct matrix inversion is rarely used, however, for solving a large system of linear equations.
