\(\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.

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