91Ó°ÊÓ

Consider the dynamical system $$\vec{x}(t+1)=\left[\begin{array}{cc} 1.1 & 0 \\ 0 & \lambda \end{array}\right] \vec{x}(t)$$ Sketch a phase portrait of this system for the given values of \(\lambda:\) $$\lambda=1.2$$

Consider a dynamical system \\[ \vec{x}(t)=\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] \\] whose transformation from time \(t\) to time \(t+1\) is given by the following equations: \\[ \begin{array}{l} x_{1}(t+1)=0.1 x_{1}(t)+0.2 x_{2}(t)+1 \\ x_{2}(t+1)=0.4 x_{1}(t)+0.3 x_{2}(t)+2 \end{array} \\] Such a system, with constant terms in the equations, is not linear, but affine. a. Find a \(2 \times 2\) matrix \(A\) and a vector \(\vec{b}\) in \(\mathbb{R}^{2}\) such that \\[ \vec{x}(t+1)=A \vec{x}(t)+\vec{b} \\] b. Introduce a new state vector \\[ \vec{y}(t)=\left[\begin{array}{c} x_{1}(t) \\ x_{2}(t) \\ 1 \end{array}\right] \\] with a "dummy" 1 in the last component. Find a \(3 \times 3\) matrix \(B\) such that \\[ \vec{y}(t+1)=B \vec{y}(t) \\] How is \(B\) related to the matrix \(A\) and the vector \(\vec{b}\) in part (a)? Can you write \(B\) as a block matrix involv\(\operatorname{ing} A\) and \(\vec{b} ?\) c. What is the relationship between the eigenvalues of \(A\) and \(B ?\) What about eigenvectors? d. For arbitrary values of \(x_{1}(0)\) and \(x_{2}(0),\) what can you say about the long-term behavior of \(x_{1}(t)\) and \(x_{2}(t) ?\)

Give an example of a \(4 \times 4\) matrix \(A\) without real eigenvalues.

Do there exist invertible \(n \times n\) matrices \(A\) and \(B\) such that \(A B-B A=A\) ? Explain.

In his groundbreaking text Ars Magna (Nuremberg, 1545 ), the Italian mathematician Gerolamo Cardano explains how to solve cubic equations. In Chapter XI, he considers the following example: \\[x^{3}+6 x=20.\\] a. Explain why this equation has exactly one (real) solution. Here, this solution is easy to find by inspection. The point of the exercise is to show a systematic way to find it. b. Cardano explains his method as follows (we are using modern notation for the variables): "I take two cubes \(v^{3}\) and \(u^{3}\) whose difference shall be \(20,\) so that the product \(v u\) shall be \(2,\) that is, a third of the coefficient of the unknown \(x\). Then, I say that \(v-u\) is the value of the unknown \(x \cdot "\) Show that if \(v\) and \(u\) are chosen as stated by Cardano, then \(x=v-u\) is indeed the solution of the equation \(x^{3}+6 x=20\). c. Solve the system \\[\left|\begin{array}{r} v^{3}-u^{3}=20 \\ v u=2 \end{array}\right|\\] to find \(u\) and \(v\) d. Consider the equation \\[x^{3}+p x=q,\\] where \(p\) is positive. Using your work in parts (a), (b), and (c) as a guide, show that the unique solution of this equation is \\[\begin{aligned} x=& \sqrt[3]{\frac{q}{2}+\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}} \\\ &-\sqrt[3]{-\frac{q}{2}+\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}}. \end{aligned}\\] This solution can also be written as \\[x=\sqrt[3]{\frac{q}{2}+\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}}\\] \\[+\sqrt[3]{\frac{q}{2}-\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}}.\\] What can go wrong when \(p\) is negative? e. Consider an arbitrary cubic equation \\[x^{3}+a x^{2}+b x+c=0.\\] Show that the substitution \(x=t-(a / 3)\) allows you to write this equation as \\[t^{3}+p t=q.\\]

Find an eigenbasis for each of the matrices \(A\) in Exercises 50 through \(54,\) and thus diagonalize A. Hint: Exercise 48 is helpful. $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{array}\right]$$