# Archive | courses

## ECE 515, 9.11

Lecture notes, 3.1-3.5.

• Cayley-Hamilton Theorem;
• Matrix exponentials;
• Solutions to linear systems differential equations.

## Math 487, 9.10

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

• Vandermonde determinants.
• Applications for Lagrange interpolation.

## 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
$$## ECE 515, Homework 1 $$\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}}$$ 1. (30) Consider the dynamical system$$
\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)$$?. 4. Linearize the ## Math 487, 9.7 Continuing to use Olver’s book, chapter 1, but also relying on Prasolov’s notes on linear algebra. • Determinants and $$LU$$ decomposition. • Determinants of products of matrices. • Kramer’s rule. • Useful determinants: • Tridiagonal matrices. ## ECE 515, 9.6 Following lecture notes, 2.6, 2.7: • Eigenvalues and eigenvectors; • Operators with distinct eigenvalues; diagonalization; • Failure to diagonalize leads to considering chains of subspaces $$V^{(k)}_\lambda:={\mathtt{Ker}} (A_\lambda)^k, A_\lambda:=A-\lambda E$$, which lead to • Jordan normal form. ## Math 487, 9.5 Continuing to use Olver’s book, chapter 1, but also relying on Prasolov’s notes on linear algebra. • Determinant (as a function from $${\mathtt{Mat}}(n\times n;\mathbf{k})$$: existence and properties. • Matrix is singular iff its determinant vanishes. ## ECE 515, 9.4 $$\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}}$$ • Bra-Ket notation It is useful to use the notion of linear functions on a linear space (what is traditionally represented by vector-rows). One can multiply vector row by vector-column to get a number (an element of the base ## Math 487, 8.31 Still on Olver’s book, ch. 1. Notion of (matrix) group: axioms – closedness, associativity, existence of unity, inverse. Examples: groups of lower, upper triangular matrices. Matrix inverses. Row swaps and permutation matrices. Group of permutation matrices. Non-singular matrices and … ## Math 487, 8.29 • Interlude: idempotent rings and optimization problems • Gaussian elimination and lower triangular matrices. Practice exercise: 1. Consider the network with the cost of moving between the cities given by the matrix$$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