\(\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.

Dear Prof.

is it possible for you to point to some references that give a more rigorous introduction and explanation of what you talked in class ? Apparently I think you are not following the Course Notes (by Basar etc), nor the Haspenda book. Some of the concepts you mentioned (like Jordan normal form, cyclic vectors) couldn’t not be found (or merely just mentioned) in the notes and and the book. References with relevant examples/exercises will be really helpful. Thanks !

Browse any linear algebra textbook. I personally like Gelfand’s Lectures on Linear Algebra; this one is available online (for UofI students) and is not bad…