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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 …

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september 25

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

Lyapunov’s direct method, cont’d

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous Lyapunov function for a linear system remains one after a perturbation of the system.

Examples.

Good homogeneous functions: quadratic forms.

Quadratic forms

Coordinate forms. …

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ece515: homework 2

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of september 25.

  1. Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\), and the operator \(D:V\to V\) of differentiation. Compute
    \[
    \exp(tD).
    \]
  2. Let
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september 18

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

Remarks
  • What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
  • For LTVs, any operator \(\Phi\) with \(\det(\Phi)>0\) can be a fundamental solution; for LTIs the situation is far more complicated (say, the

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september 16

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

linear time dependent systems

Generalizing the notion of state transition matrix, we can address also the time-dependent systems
\[
\dot{x}(t)=A(t)x(t).
\]
Here, we dove the equation on \([t_1,t_2]\) with \(x(t_1)=x\); the value of solution at …

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september 11

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

matrix exponential

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

Computing the matrix exponentials can be done by just taking power series (not advised),

or by using the Jordan normal forms…

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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 …

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ece515, homework1 (due 9.11).

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\). There exists a natural basis in \(V\), comprised of monomials – we will denote this basis as \(\e\)).

  1. Is
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september 4

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

Linear operators

Their range=image, null-space=kernel (both linear subspaces).

Rank=dimension of the image (if finite-dimensional).

Important: dimension of the kernel + rank=dimension of the domain…

Example:

\(V=\Fun(S,\Field)\), where, as before \(S=\{0,1,2,3,4,5\}\subset\Comp\). Let \(Z\) is the operator …

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september 2

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

Fields, vector spaces, subspaces, linear operators, range space, null space

A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.

Examples (Common fields)
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