Eigenvalues and Eigenvectors

The polynomial $p(\lambda) = {\rm det}(\lambda I-A)$ is called the characteristic polynomial of $A$. The roots of $p(\lambda)=0$ are called the eigenvalues of $A$. Since the degree of $p(\lambda)$ is $n$, it has $n$ roots, and so $A$ has $n$ eigenvalues. Eigenvalues of a real matrix may be real or appear in complex pairs.

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

An $n$ by $n$ matrix need not have $n$ independent eigenvectors. The simplest example is

$$A (\epsilon) = \bmat{cc} 0 & 1 \\ \epsilon^2 & 0 \emat,$$ (3)

whose eigenvalues are $\lambda_{\pm} = \pm \epsilon$. When $\epsilon \neq 0$, there are two right eigenvectors, $x_{\pm} = [ 1, \pm \epsilon ]^T$. As $\epsilon$ approaches 0, the two eigenvectors approach one another. When $\epsilon = 0$, both eigenvalues equal 0, and there is a single independent right eigenvector parallel to $[1,0]^T$. 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 NHEP.

Since the eigenvalues may be complex, 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).