Linear operators; normal forms
Linear operators between different spaces are easy to normalize: the rank is the only invariant; the classification is discrete.
For endomorphisms, the situation is more involved.
Diagonalization corresponds to splitting the space into the simplest pieces, 1-dimensional. It is not always possible.
Example: differentiation in the basis \(x^k/k!\).
In this case – apply Jordan normal form
Jordan blocks (example: differentiating quasi polynomials). Structure theorem.
Other normal forms: cyclic vectors…
Other classification problems (quivers…)
Powers of an \(n\times n\) matrix sit in \(n^2\)-dimensional space; there is a lot of room – but \(n+1\) of them are already linearly dependent!
Also helps find functions of matrices, such as
which is instrumental in finding solutions to linear systems of differential equations.
Computing the matrix exponentials.