Eigenvalues and Eigenvectors

The polynomial $p(\lambda) = {\rm det}(\lambda B-A)$ is called the characteristic polynomial of $A - \lambda B$. The roots of $p(\lambda)=0$ are eigenvalues of $A - \lambda B$. Since the degree of $p(\lambda)$ is $n$, it has $n$ roots, and so $A - \lambda B$ has $n$ eigenvalues.

A nonzero vector $x$ satisfying $Ax = \lambda Bx$ is a (right) eigenvector for the eigenvalue $\lambda$. The eigenpair $(\lambda,x)$ also satisfies $x^* A = \lambda x^* B$, so we can also call $x$ a left eigenvector.

All eigenvalues of the definite pencil $A - \lambda B$ are real. This lets us write them in sorted order $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. If all $\lambda_i > 0$, then $A - \lambda B$ is called positive definite, and if all $\lambda_i \geq 0$, then $A - \lambda B$ is called positive semidefinite. Negative definite and negative semidefinite are defined analogously. If there are both positive and negative eigenvalues, $A - \lambda B$ is called indefinite.

Each $x_i$ is real if $A$ and $B$ are real. Though the $x_i$ may not be unique, they may be chosen to all be $B$ orthogonal to one another: $x_i^* B x_j=0$ if $i \neq j$. This is also called orthogonality with respect to the inner product induced by the Hermitian positive definite matrix $B$. When an eigenvalue is distinct from all the other eigenvalues, its eigenvector is unique (up to multiplication by scalars).