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 their matrices.

  • Inner products, norms,
  • symmetric matrices, symmetric and antisymmetric parts of a matrix,
  • quadratic forms,

2 Responses to ECE 515, 9.18

  1. Heng-Sheng Chang September 19, 2018 at 1:58 pm #

    Course notes about the relation between det $G$ and the set of independent vectors $\{v_i\}_{i=1}^n$

    Note that $G$ is the Gramian matrix $G=V^TV$, and $V=[v_1\ v_2\ \dots\ v_n]$.\\

    $\{v_i\}$ is linear dependent\\
    $\iff$ there exists a vector $c\neq0$ s.t. $\sum c_iv_i=0$\\
    $\iff\ \exists\ c\neq0$ s.t. $=0\ \forall j$\\
    $\iff\ \exists\ c\neq0$ s.t. $\sum c_i=0\ \forall j$\\
    $\iff\ \exists\ c\neq0$ s.t. $Gc=0$\\
    $\iff$ det $G=0$

    • Heng-Sheng Chang September 19, 2018 at 2:25 pm #

      not sure how to use this site… how to get a displayed equation?

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