Linear Systems Theory

If you are interested in learning more about operator norms, please review Lecture 7 in the 510 lecture notes. The basic idea is that a matrix is simply a linear operator, and hence the metric on our underlying Hilbert space induces a norm on that operator. For example, consider operator

$A:X\to Y$

The operator norm is just

$\begin{aligned} \|A\|_{\text{op}}&=\sup_{\|x\|=1}\|Ax\|\\ &=\sup\left\{\frac{\|Ax\|}{\|x\|}:\ \ x\neq 0, \ x\in X\right\} \end{aligned}$

For example, if $A$ has as its domain $(\mathbb{R}^n,\|\cdot\|_2)$ and as its co-domain $(\mathbb{R}^n,\|\cdot\|_2)$, then the operator norm is simply the induced 2-norm or maximum singular value. If $A$ has as its domain $(\mathbb{R}^n,\|\cdot\|_1)$ and as its co-domain $(\mathbb{R}^n,\|\cdot\|_1)$, then the operator norm is simply the induced 1-norm or maximum $\ell_1$-norm of the columns.

One can also mix and match norms on domain and co-domain. For example, if $A$ has as its domain $(\mathbb{R}^n,\|\cdot\|_1)$ and as its co-domain $(\mathbb{R}^n,\|\cdot\|_2)$, then the operator norm is simply the maximum across the column’s $\ell_2$-norms.

This is a recommendation on how to watch M2-RL2. This video is ~1hour long. It has three concise parts however.

• Part 1: 0 – 17.5mins: Introduction to spectral theorems on stability
• Part 2: 17.5 – 39.5mins: Numerical integration and connections between DT and CT stability spectral properties.
• Part 3: 39.5 – 59.5mins: Linearization of nonlinear systems and stability via Hartman-Grobman.

You can break this up into roughly three equal parts and watch them this way.

Homework 1 has been posted. It is due 2022-01-16 at 11:59PM.