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 a2cos(2ϕ)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 AA be an n×mn\times m matrix, then we define an eigenvalue λ\lambda and a corresponding eigenvector v as a scalar and a vector such that

A𝒗=λ𝒗.A\textbf{v}=\lambda\textbf{v}. (19.138)
Theorem 19.5.1

The scalar λ\lambda is an eigenvector of AA if and only if

det(AλI)=0\det(A-\lambda I)=0 (19.139)

Let λ\lambda be such that

det(AλI)=0.\det(A-\lambda I)=0. (19.140)

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

dim(N(AλI))>0.\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 𝐯0\mathbf{v}\neq 0 such that (AλI)𝐯=0(A-\lambda I)\mathbf{v}=0, or equivalently that

A𝐯\displaystyle A\mathbf{v} =λI𝐯\displaystyle=\lambda I\mathbf{v} (19.142)
=λ𝐯\displaystyle=\lambda\mathbf{v} (19.143)

Thus we have proven both directions.


Definition 19.5.2

We define the \sayeigenspace of an eigenvalue to be the complete set of all corresponding eigenvalues, that is, we write

Eλ={𝐯:A𝐯=λ𝐯}.E_{\lambda}=\{\mathbf{v}:A\mathbf{v}=\lambda\mathbf{v}\}. (19.144)