# september 30

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

#### Lyapunov’s direct method, cont’d

For complex spaces quadratic forms are not really suitable (if one wants just a real number as a result, the signatures are all $$(n,n)$$

Sylvester criterion for positive definiteness.
(Bonus: Rayleigh characterization of eigenvalues.)

#### Interlude: complex eigenvalues and eigenvectors.

Jordan normal forms in real vector spaces.

#### Quadratic forms as Lyapunov functions

If the operator defines an asymptotically stable system, it is Hurwitz. For a Hurwitz operator, a quadratic form exists which is a (strict) Lyapunov function. A strict Lyapunov function implies AS.

Lyapunov equation.

Discrete time systems.

#### Bounded Input Bounded Output

GES implies BIBO; not vice versa.