Quadratic Eigenvalue Problem - Methods of Solution

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


L(\lambda) =
\lambda
\begin{bmatrix}
M & 0 \\
0 & I_n
\end{bmatrix}
+
\begin{bmatrix}
C & K \\
-I_n & 0
\end{bmatrix},

where is the -by- identity matrix, with corresponding eigenvector


z =
\begin{bmatrix}
\lambda x \\
x
\end{bmatrix}.

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:

    A woman might claim to retain some of the child’s faculties, although very limited and defused, simply because she has not been encouraged to learn methods of thought and develop a disciplined mind. As long as education remains largely induction ignorance will retain these advantages over learning and it is time that women impudently put them to work.
    Germaine Greer (b. 1939)

    The reading public is intellectually adolescent at best, and it is obvious that what is called “significant literature” will only be sold to this public by exactly the same methods as are used to sell it toothpaste, cathartics and automobiles.
    Raymond Chandler (1888–1959)

    Any solution to a problem changes the problem.
    —R.W. (Richard William)