I am looking for solution algorithms for a second order generalization of the eigenvalue problem.
A, B, and C are n-by-n matrices, I is the n-dimensional identity matrix, $\lambda_i$ is an unknown scalar, and $v_i$ is an unknown vector. The eigenvalue problem can be stated as
- $(\lambda_i I-A)v_i=0\longrightarrow (\lambda_i,v_i)$
The generalized eigenvalue problem (every search I have tried points me to this problem) is then
- $(\lambda_i B-A)v_i=0\longrightarrow (\lambda_i,v_i)$
I would like some guidance for a second order generalization of the eigenvalue problem.
- $(\lambda_i^2C+\lambda_i B-A)v_i=0\longrightarrow (\lambda_i,v_i)$
Does this problem already have a name? What key words will lead me to more information on this problem? Solution algorithms avoiding calculations of determinants for Eq. 2 have implementations such as Mathematica's Eigensystem command. Do similar implementations exist for Eq. 3 (not necessarily in Mathematica)?
What you are looking for is the polynomial eigenvalue problem which considers finding scalars $\lambda$ and nonzero vectors $v$ such that $$ \left(\sum_{i=0}^n \lambda^i A_i\right) v = 0, $$ where $A_0,\ldots,A_n$ are given square matrices.
For $n=1$ and $A_0=I$, one gets the "ordinary" eigenvalue problem while $A_0\neq I$ leads to the so-called generalized eigenvalue problem. With $n=2$, we get the quadratic eigenvalue problem $$ (\lambda^2 A_2 +\lambda A_1 + A_0)v = 0. $$
The polynomial eigen-solver is implemented, e.g., in the
polyeigfunction in MATLAB.