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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

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

  1. (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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ECE 515, 9.6

Following lecture notes, 2.6, 2.7:

  • Eigenvalues and eigenvectors;
  • Operators with distinct eigenvalues; diagonalization;
  • Failure to diagonalize leads to considering chains of subspaces \(V^{(k)}_\lambda:={\mathtt{Ker}} (A_\lambda)^k, A_\lambda:=A-\lambda E\), which lead to
  • Jordan normal form.
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Math 487, 9.5

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

  • Determinant (as a function from \({\mathtt{Mat}}(n\times n;\mathbf{k})\): existence and properties.
  • Matrix is singular iff its determinant vanishes.
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ECE 515, 9.4

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

  • Bra-Ket notation
    It is useful to use the notion of linear functions on a linear space (what is traditionally represented by vector-rows). One can multiply vector row by vector-column to get a number (an element of the base
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Math 487, 8.31

Still on Olver’s book, ch. 1.
Notion of (matrix) group: axioms – closedness, associativity, existence of unity, inverse.
Examples: groups of lower, upper triangular matrices. Matrix inverses.
Row swaps and permutation matrices. Group of permutation matrices.
Non-singular matrices and …

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Math 487, 8.29

  • Interlude: idempotent rings and optimization problems
  • Gaussian elimination and lower triangular matrices.

Practice exercise:

    1. Consider the network with the cost of moving between the cities given by the matrix
      $$A=\left\vert
      \begin{array}{cccc}0 & 10&25&35\\
      10&0&10&\infty\\
      25&10&0&10\\
      35&\infty&10&0\\
      \end{array}
      \right\vert
      $$
      Using
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