Methods of Solution
Direct methods for solving the standard or generalized eigenvalue problems and are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials. One approach is to transform the quadratic matrix polynomial to a linear matrix pencil, and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
The most common linearization is the first companion linearization
where is the -by- identity matrix, with corresponding eigenvector
We solve for and, for example by computing the Generalized Schur form. We can then take the first components of as the eigenvector of the original quadratic .
Read more about this topic: Quadratic Eigenvalue Problem
Famous quotes containing the words methods of, methods and/or solution:
“I believe in women; and in their right to their own best possibilities in every department of life. I believe that the methods of dress practiced among women are a marked hindrance to the realization of these possibilities, and should be scorned or persuaded out of society.”
—Elizabeth Stuart Phelps (1844–1911)
“It would be some advantage to live a primitive and frontier life, though in the midst of an outward civilization, if only to learn what are the gross necessaries of life and what methods have been taken to obtain them.”
—Henry David Thoreau (1817–1862)
“The Settlement ... is an experimental effort to aid in the solution of the social and industrial problems which are engendered by the modern conditions of life in a great city. It insists that these problems are not confined to any one portion of the city. It is an attempt to relieve, at the same time, the overaccumulation at one end of society and the destitution at the other ...”
—Jane Addams (1860–1935)