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ECE 515, 9.18

Covering course notes, 3.6-8 and 2-8-10.

  • Peano-Baker series for fundamental matrices, – properties, convergence, role of commutativity;
  • Solutions for forced systems for LTV systems;
  • General classification of spaces of solutions of LTI;

Quadratic and Hermitian forms, norms, self-adjoint operators and …

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ECE 515, Homework 1, solutions.

\(\def\Real{\mathbb{R}}\def\ee{\mathbf{e}}\def\ff{\mathbf{f}}\newcommand\ket[1]{\vert{#1}\rangle}\newcommand\bra[1]{\langle{#1}\vert}\newcommand\bk[2]{\langle #1\vert#2\rangle}\def\lin{\mathtt{Lin}} \)

  1. (30) Consider the dynamical system
    $$
    \dot{x}=x-(x+y)^2/4;\\
    \dot{y}=y-(x+y)^2/4.
    $$
    1. Find equilibrium points.
      Setting right hand sides to zero, we obtain that \(x=y\), and, consequently, \(x=x^2\). Hence the equilibria are \((0,0)\) and \((1,1)\).
    2. Linearize the system near the
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Math 487, 9.14

Material from chapter 9 of Stanley’s book.

  • Graphs and their Laplacians
  • Spanning trees
  • Matrix-tree theorem
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ECE 515, 9.13

Lecture notes, 3.1-3.5.

  • Solutions of non-homogenous linear systems;
  • Exponentials of diagonal and diagonalizable operators and matrices;
  • Lagrange approximations and reduction of matrix functions to polynomials;
  • Non-autonomous systems and Picard approximations.
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Math 487, 9.12

Still on Prasolov’s notes on linear algebra, chapter 1.

  • Circulant matrices and determinants.
  • Minors and applications.
    • Definitions. Cofactors.
    • Laplace expansion.
    • Binet-Cauchy theorem.
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ECE 515, 9.11

Lecture notes, 3.1-3.5.

  • Cayley-Hamilton Theorem;
  • Matrix exponentials;
  • Solutions to linear systems differential equations.
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Math 487, 9.10

Mostly following Prasolov’s notes on linear algebra, chapter 1.

  • Vandermonde determinants.
  • Applications for Lagrange interpolation.
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Math 487, Homework 1

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  1. (15) Find all matrices
    $$
    X=\left(
    \begin{array}{cc}
    x&y\\
    z&w
    \end{array}
    \right)
    $$
    such that \(XA=BX\), where
    $$
    A=\left(
    \begin{array}{cc}
    1&2\\
    -1&0
    \end{array}
    \right), \mathrm{\ and\ }
    B=\left(
    \begin{array}{cc}
    0&1\\
    3&0
    \end{array}
    \right).
    $$
  2. (15) Same for
    $$
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ECE 515, Homework 1

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  1. (30) Consider the dynamical system
    $$
    \dot{x}=x-(x+y)^2/4;\\
    \dot{y}=y-(x+y)^2/4.
    $$
    1. Find equilibrium points.
    2. Linearize the system near the corresponding constant solutions.
    3. Find a trajectory connecting some of those equilibrium points. Hint: what happens if \(x(0)=y(0)\)?.
    4. Linearize the
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Math 487, 9.7

Continuing to use Olver’s book, chapter 1, but also relying on Prasolov’s notes on linear algebra.

  • Determinants and \(LU\) decomposition.
  • Determinants of  products of matrices.
  • Kramer’s rule.
  • Useful determinants:
    • Tridiagonal matrices.
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