Eigenvalues and Eigenvectors

The polynomial $p(\lambda) = {\rm det}(\lambda B-A)$ is the characteristic polynomial of $A - \lambda B$. The degree of $p(\lambda)$ is at most $n$. The roots of $p(\lambda)=0$ are called the finite eigenvalues of $A - \lambda B$. If the degree of $p(\lambda)$ is $d<n$, we say that $A - \lambda B$ has $n-d$ infinite eigenvalues too. For example,

$$A - \lambda B = \bmat{ccc} 1 & & \\ & 1 & \\ & & 0 \emat - \lambda \bmat{ccc} 2 & & \\ & 0 & \\ & & 1 \emat$$

has characteristic polynomial $p( \lambda ) = (1 - 2 \cdot \lambda)(1 - 0 \cdot \lambda)(0 - 1 \cdot \lambda) = 2 \lambda^2 - \lambda$, and so has eigenvalues $.5$, $\infty$, and $0$.

If $\lambda$ is a finite eigenvalue, a nonzero vector $x$ satisfying $Ax = \lambda Bx$ is a (right) eigenvector for the eigenvalue $\lambda$. A nonzero vector $y$ satisfying $y^* A = \lambda ^* B$ is a left eigenvector.

If $\infty$ is an eigenvalue, nonzero vectors $x$ and $y$ satisfying $Bx=0$ and $y^*B = 0$ are called right and left eigenvectors, respectively.

An $n$ by $n$ pencil $A - \lambda B$ need not have $n$ independent eigenvectors. The simplest example is $A( \epsilon ) - \lambda I$, which is defined in equation (2.3) and discussed in §2.5.1. The fact that $n$ independent eigenvectors may not exist (though there is at least one for each distinct eigenvalue) will necessarily complicate both theory and algorithms for the GNHEP.

Since the eigenvalues may be complex or infinite, there is no fixed way to order them. Nonetheless, it is convenient to number them as $\lambda_1 ,\ldots,\lambda_n$, with corresponding right eigenvectors $x_1,\ldots,x_n$ and left eigenvectors $y_1,\ldots,y_n$ (if they exist).