## Subsection9.2.2Singular matrices and the eigenvalue problem

###### Definition9.2.2.1. Eigenvalue, eigenvector, and eigenpair.

Let $A \in \C^{m \times m} \text{.}$ Then $\lambda \in \C$ and nonzero $x \in \C^{m}$ are said to be an eigenvalue and corresponding eigenvector if $A x = \lambda x \text{.}$ The tuple $( \lambda, x )$ is said to be an eigenpair.

$A x = \lambda x$ means that the action of $A$ on an eigenvector $x$ is as if it were multiplied by a scalar. The direction does not change: only its length is scaled:

As part of an introductory course on linear algebra, you learned that the following statements regarding $m \times m$ matrix $A$ are all equivalent:

• $A$ is nonsingular.

• $A$ has linearly independent columns.

• There does not exists $x \neq 0$ such that $A x = 0 \text{.}$

• $\Null( A ) = \{ 0 \} \text{.}$ (The null space of $A$ is trivial.)

• $\dim(\Null( A )) = 0 \text{.}$

• $\det( A ) \neq 0 \text{.}$

Since $A x = \lambda x$ can be rewritten as $( \lambda I - A ) x = 0 \text{,}$ we note that the following statements are equivalent for a given $m \times m$ matrix $A \text{:}$

• $( \lambda I - A )$ is singular.

• $( \lambda I - A )$ has linearly dependent columns.

• There exists $x \neq 0$ such that $( \lambda I - A ) x = 0 \text{.}$

• The null space of $\lambda I - A$ is nontrivial.

• $\dim(\Null( \lambda I - A )) \gt 0 \text{.}$

• $\det( \lambda I - A ) = 0 \text{.}$

It will become important in our discussions to pick the right equivalent statement.

###### Definition9.2.2.2. Spectrum of a matrix.

The set of all eigenvalues of $A$ is denoted by $\Lambda( A )$ and is called the spectrum of $A \text{.}$

The spectral radius of $A \text{,}$ $\rho( A ) \text{,}$ equals the magnitude of the largest eigenvalue, in magnitude: