Exercise7
This exericse concerns the matrix transformations defined by matrices of the form
\begin{equation*} A = \left[\begin{array}{rr} a \amp b \\ b \amp a \\ \end{array}\right] \text{.} \end{equation*}Let's begin by looking at two special types of these matrices.

First, consider the matrix where \(a = 2\) and \(b=0\) so that
\begin{equation*} A = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 2 \\ \end{array}\right] \text{.} \end{equation*}Describe the geometric effect of this matrix. More generally, suppose we have
\begin{equation*} A = \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \text{,} \end{equation*}where \(r\) is a positive number. What is the geometric effort of \(A\) on vectors in the plane?

Suppose now that \(a = 0\) and \(b = 1\) so that
\begin{equation*} A = \left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right] \text{.} \end{equation*}What is the geometric effect of \(A\) on vectors in the plane? More generally, suppose we have
\begin{equation*} A = \left[\begin{array}{rr} \cos\theta \amp \sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*}What is the geometric effect of \(A\) on vectors in the plane?

In general, the composition of matrix transformation depends on the order in which we compose them. For these transformations, however, it is not the case. Check this by verifying that
\begin{equation*} \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta \amp \sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] = \left[\begin{array}{rr} \cos\theta \amp \sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \text{.} \end{equation*} 
Let's now look at the general case where
\begin{equation*} A = \left[\begin{array}{rr} a \amp b \\ b \amp a \\ \end{array}\right] \text{.} \end{equation*}We will draw the vector \(\twovec{a}{b}\) in the plane and express it using polar coordinates \(r\) and \(\theta\) as shown in FigureĀ 21.
We then have
\begin{equation*} \twovec{a}{b} = \twovec{r\cos\theta}{r\sin\theta} \text{.} \end{equation*}Show that the matrix
\begin{equation*} \left[\begin{array}{rr} a \amp b \\ b \amp a \\ \end{array}\right] = \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta \amp \sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*} 
Using this description, describe the geometric effect on vectors in the plane of the matrix transformation defined by
\begin{equation*} A= \left[\begin{array}{rr} a \amp b \\ b \amp a \\ \end{array}\right] \text{.} \end{equation*} 
Suppose we have a matrix transformation \(T\) defined by a matrix \(A\) and another transformation \(S\) defined by \(B\) where
\begin{equation*} A= \left[\begin{array}{rr} a \amp b \\ b \amp a \\ \end{array}\right], B= \left[\begin{array}{rr} c \amp d \\ d \amp c \\ \end{array}\right] \text{.} \end{equation*}Describe the geometric effect of the composition \(S\circ T\) in terms of the \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\)
The matrices of this form give a model for the complex numbers and will play an important role in <<seccomplexeigenvalues>>.