# A calculus of the absurd

#### 20.7 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 Love and Tensor Algebra 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).

##### 20.7.1 Eigenvalues and eigenvectors
• Definition 20.7.1 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

$$A\textbf {v} = \lambda \textbf {v}.$$

• Theorem 20.7.1 The scalar $$\lambda$$ is an eigenvector of $$A$$ if and only if

$$\det (A-\lambda I) = 0$$

Let $$\lambda$$ be such that

$$\det (A - \lambda I) = 0.$$

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

$$\dim \left (N(A - \lambda I )\right ) > 0.$$

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

$$E_{\lambda } = \{\mathbf {v} : A \mathbf {v} = \lambda \mathbf {v}\}.$$