19.5 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 \sayhere be dragons-style are some subtleties (this might also be me trying to violate the pigeonhole principle with my personal timetabling).

19.5.1 Eigenvalues and eigenvectors

Definition 19.5.1

Let $A$ be an $n\times m$ matrix, then we define an eigenvalue $\lambda$ and a corresponding eigenvector v as a scalar and a vector such that

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

(19.138)

Theorem 19.5.1

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

\det(A-\lambda I)=0

(19.139)

Let $\lambda$ be such that

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

(19.140)

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

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

(19.141)

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

	$\displaystyle A\mathbf{v}$	$\displaystyle=\lambda I\mathbf{v}$		(19.142)
		$\displaystyle=\lambda\mathbf{v}$		(19.143)

Thus we have proven both directions.

$\Box$

Definition 19.5.2

We define the \sayeigenspace 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}\}.

(19.144)