Chapter 5.2, Problem 14E

Interpretation Introduction

Interpretation:

To find the probabilities of all the different kinds of fixed points of the system $\dot{x}=Ax$, where $\text{A=}\left(\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right)$. Suppose, we pick the entries $\text{a, b,c, d}$ independently and at random, form a uniform distribution on the interval $\left[-1,1\right]$

Concept Introduction:

$\text{ Av = λv}$, the desired straight line solution exists, if v is an eigen vector of A with corresponding eigen value $\text{λ}$

In general, the eigen value of a matrix A is given by the characteristic equation

$\text{det}\left(\text{A-λI}\right)\text{ = 0}$, where I is the identity matrix