# september 9

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$

#### 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…)

#### Cayley-Hamilton Theorem

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

#### matrix exponential

which is instrumental in finding solutions to linear systems of differential equations.

Computing the matrix exponentials.