A calculus of the absurd

22.6 Spectral theory

I see the eigenvalue in thine eye, I hear the tender tensor in thy sigh. Bernoulli would have been content to die, Had he but known such \(a^2 \cos (2 \phi )\)!

— Stanislaw Lem L OVE AND T ENSOR A LGEBRA from the Cyberiad

Spectral theory is one of those things which is at once kind of obvious, lurking beneath the covers “here be dragons”-style are some subtleties (this might also be me trying to violate the pigeonhole principle with my personal timetabling).

22.6.1 Eigenvalues and eigenvectors

  • Definition 22.6.1 Let \(A\) be an \(n \times m\) matrix, then we define an eigenvalue \(\lambda \) and a corresponding eigenvector \(\textbf {v}\) as a scalar and a vector such that

    \begin{equation} A\textbf {v} = \lambda \textbf {v}. \end{equation}

  • Theorem 22.6.1 The scalar \(\lambda \) is an eigenvector of \(A\) if and only if

    \begin{equation} \det (A-\lambda I) = 0 \end{equation}

Let \(\lambda \) be such that

\begin{equation} \det (A - \lambda I) = 0. \end{equation}

This is if and only if \(A - \lambda I\) is singular, i.e. if

\begin{equation} \dim \left (N(A - \lambda I )\right ) > 0. \end{equation}

This is equivalent (i.e. if and only if) to the statement that there exists some \(\mathbf {v} \ne 0\) such that \((A - \lambda I) \mathbf {v} = 0\), or equivalently that

\begin{align} A \mathbf {v} &= \lambda I \mathbf {v} \\ &= \lambda \mathbf {v} \end{align}

Thus we have proven both directions.

\(\Box \)

  • Definition 22.6.2 We define the “eigenspace” of an eigenvalue to be the complete set of all corresponding eigenvalues, that is, we write

    \begin{equation} E_{\lambda } = \{\mathbf {v} : A \mathbf {v} = \lambda \mathbf {v}\}. \end{equation}