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 !
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
Let be an matrix, then we define an eigenvalue and a corresponding eigenvector v as a scalar and a vector such that
The scalar is an eigenvector of if and only if
Let be such that
This is if and only if is singular, i.e. if
This is equivalent (i.e. if and only if) to the statement that there exists some such that , or equivalently that
Thus we have proven both directions.
We define the \sayeigenspace of an eigenvalue to be the complete set of all corresponding eigenvalues, that is, we write