Skip to main content

Subsection 9.2.2 Singular matrices and the eigenvalue problem

Definition 9.2.2.1. 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.

Definition 9.2.2.2. Spectrum of a matrix.

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

Definition 9.2.2.3. Spectral radius.

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

\begin{equation*} \rho( A ) = \max_{\lambda \in \Lambda( A )} \vert \lambda \vert. \end{equation*}