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.

A nonzero vector $x$ satisfying $Ax = \lambda x$ is a (right) eigenvector for the eigenvalue $\lambda$. Since $x^{\ast} A = \lambda x^{\ast}$, left and right eigenvectors are identical.

All eigenvalues of a Hermitian matrix $A$ 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 $A$ is called positive definite, and if all $\lambda_i \geq 0$, then $A$ is called positive semidefinite. Negative definite and negative semidefinite are defined analogously. If there are both positive and negative eigenvalues, $A$ is called indefinite.

Each eigenvalue $\lambda_i$ has an eigenvector $x_i$. We may assume without loss of generality that $\Vert x_i\Vert _2=1$. Each $x_i$ is real if $A$ is real. Though the $x_i$ may not be unique (e.g., any vector is an eigenvector of the identity matrix), they may be chosen to all be orthogonal to one another: $x_i^*x_j=0$ if $i \neq j$. When an eigenvalue is distinct from all the other eigenvalues, its eigenvector is unique (up to multiplication by scalars).