Matrix Mappings

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Having illustrated the notational advantage of using matrices and matrix multiplication, we now begin to discuss why there is also a conceptual advantage to matrix multiplication, a conceptual advantage that will help us to understand how systems of linear equations and linear differential equations may be solved.

Matrix multiplication allows us to view $m\times n$ matrices as mappings from $\R ^n$ to $\R ^m$ . Let $A$ be an $m\times n$ matrix and let $x$ be an $n$ vector. Then $x \mapsto Ax$ defines a mapping from $\R ^n$ to $\R ^m$ .

The simplest example of a matrix mapping is given by $1\times 1$ matrices. Matrix mappings defined from $\R \to \R$ are $x \mapsto ax$ where $a$ is a real number. Note that the graph of this function is just a straight line through the origin (with slope $a$ ). From this example we see that matrix mappings are very special mappings indeed. In higher dimensions, matrix mappings provide a richer set of mappings; we explore here planar mappings — mappings of the plane into itself — using MATLAB graphics and the program map.

The simplest planar matrix mappings are the dilatations. Let $A=cI_2$ where $c>0$ is a scalar. When $c<1$ vectors are contracted by a factor of $c$ and and these mappings are examples of contractions. When $c>1$ vectors are stretched or expanded by a factor of $c$ and these dilatations are examples of expansions. We now explore some more complicated planar matrix mappings.

The next planar motions that we study are those given by the matrices $A=\mattwo {\lambda }{0}{0}{\mu }.$ Here the matrix mapping is given by $(x,y)\mapsto (\lambda x,\mu y)$ ; that is, a mapping that independently stretches and/or contracts the $x$ and $y$ coordinates. Even these simple looking mappings can move objects in the plane in a somewhat complicated fashion.

The Program Map

We can use MATLAB to explore planar matrix mappings in an efficient way using the program map. In MATLAB type the command

map

and a menu appears labeled MAP Setup. The $2\times 2$ matrix $\mattwo {0}{-1}{1}{0}$ has been pre-entered. Click on the Proceed button. A window entitled MAP Display appears. Click on Icons and click on an icon — say Dog. Then click in the MAP Display window and a blue ‘Dog’ will appear in that window. Now click on the Map button and a new version of the Dog will appear in yellow — but the yellow Dog is rotated about the origin counterclockwise by $90^\circ$ from the blue dog. Indeed, this matrix $A$ just rotates the plane counterclockwise by $90^\circ$ . To verify this statement just click on Map again and see that the yellow dog rotates $90^\circ$ counterclockwise into the magenta dog. Of course, the magenta dog is just rotated $180^\circ$ from the original blue dog. Clicking on Map again produces a fourth dog — this one in cyan. Finally one more click on the map button will rotate the cyan dog into a red dog that exactly covers the original blue dog.

Choose another icon from the Icons menu; a blue version of this icon appears in the MAP Display window. Now click on Map to see that your chosen icon is just rotated counterclockwise by $90^\circ$ .

Other matrices will produce different motions of the plane. You may either type the entries of a matrix in the Map Setup window and click on the Proceed button or recall one of the pre-assigned matrices listed in the menu obtained by clicking on Gallery. For example, clicking on the Contracting rotation button enters the matrix $\mattwo {0.3}{-0.8}{0.8}{0.3}$ This matrix rotates the plane through an angle of approximately $69.4^\circ$ counterclockwise and contracts the plane by a factor of approximately $0.85$ . Now click on Dog in the Icons menu to bring up the blue dog again. Repeated clicking on map rotates and contracts the dog so that dogs in a cycling set of colors slowly converge towards the origin in a spiral of dogs.

Rotations

Rotating the plane counterclockwise through an angle $\theta$ is a motion given by a matrix mapping. We show that the matrix that performs this rotation is:

$\begin{equation} \label{e:rotmat} R_\theta = \mattwo{\cos\theta}{-\sin\theta}{\sin\theta}{\cos\theta}. \end{equation}$ To verify that $R_\theta$ rotates the plane counterclockwise through angle $\theta$ , let $v_\varphi$ be the unit vector whose angle from the horizontal is $\varphi$ ; that is, $v_\varphi =(\cos \varphi ,\sin \varphi )$ . We can write every vector in $\R ^2$ as $rv_\varphi$ for some number $r\ge 0$ . Using the trigonometric identities for the cosine and sine of the sum of two angles, we have:

$\begin{eqnarray*} R_\theta (rv_\varphi) & = & \mattwo{\cos\theta}{-\sin\theta}{\sin\theta}{\cos\theta} \vectwo{r\cos\varphi}{r\sin\varphi} \\ & = & \vectwo{r\cos\theta\cos\varphi -r\sin\theta\sin\varphi} {r\sin\theta\cos\varphi + r\cos\theta\sin\varphi}\\ & = & r\vectwo{\cos(\theta+\varphi)}{\sin(\theta+\varphi)} \\ & = & rv_{\varphi+\theta}. \end{eqnarray*}$

This calculation shows that $R_\theta$ rotates every vector in the plane counterclockwise through angle $\theta$ .

It follows from (??) that $R_{180^\circ } = -I_2$ . So rotating a vector in the plane by $180^\circ$ is the same as reflecting the vector through the origin. It also follows that the movement associated with the linear map $x\mapsto -cx$ where $c>0$ may be thought of as a dilatation ( $x\mapsto cx$ ) followed by rotation through $180^\circ$ ( $x\mapsto -x$ ).

We claim that combining dilatations with general rotations produces spirals. Consider the matrix $S = \mattwo {c\cos \theta }{-c\sin \theta }{c\sin \theta }{c\cos \theta } = cR_\theta$ where $c<1$ . Then a calculation similar to the previous one shows that $S(rv_\varphi ) = c(rv_{\varphi +\theta }).$ So $S$ rotates vectors in the plane while contracting them by the factor $c$ . Thus, multiplying a vector repeatedly by $S$ spirals that vector into the origin. The example that we just considered while using map is $\mattwo {0.3}{-0.8}{0.8}{0.3}\cong \mattwo {0.85\cos (69.4^\circ )} {-0.85\sin (69.4^\circ )}{0.85\sin (69.4^\circ )}{0.85\cos (69.4^\circ )},$ which has the general form of $S$ .

A Notation for Matrix Mappings

We reinforce the idea that matrices are mappings by introducing a notation for the mapping associated with an $m\times n$ matrix $A$ . Define $L_A:\R ^n\to \R ^m$ by $L_A(x) = Ax,$ for every $x\in \R ^n$ .

There are two special matrices: the $m\times n$ zero matrix $O$ all of whose entries are $0$ and the $n\times n$ identity matrix $I_n$ whose diagonal entries are $1$ and whose off diagonal entries are $0$ . For instance, $I_3 = \left ( \begin {array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array} \right ).$

The mappings associated with these special matrices are also special. Let $x$ be an $n$ vector. Then

$\begin{equation} \label{multby0} Ox=0, \end{equation}$ where the $0$ on the right hand side of (??) is the $m$ vector all of whose entries are $0$ . The mapping $L_O$ is the zero mapping — the mapping that maps every vector $x$ to $0$ .

Similarly, $I_nx=x$ for every vector $x$ . It follows that $L_{I_n}(x) = x$ is the identity mapping, since it maps every element to itself. It is for this reason that the matrix $I_n$ is called the $n\times n$ identity matrix.

Exercises

In Exercises ?? – ?? find a nonzero vector that is mapped to the origin by the given matrix.

$A=\mattwo {0}{1}{0}{-2}$ .

$B=\mattwo {1}{2}{-2}{-4}$ .

$C=\mattwo {3}{-1}{-6}{2}$ .

What $2\times 2$ matrix rotates the plane about the origin counterclockwise by $30^\circ$ ?

What $2\times 2$ matrix rotates the plane clockwise by $45^\circ$ ?

What $2\times 2$ matrix rotates the plane clockwise by $90^\circ$ while dilating it by a factor of $2$ ?

Find a $2\times 2$ matrix that reflects vectors in the $(x,y)$ plane across the $x$ axis.

Find a $2\times 2$ matrix that reflects vectors in the $(x,y)$ plane across the $y$ axis.

Find a $2\times 2$ matrix that reflects vectors in the $(x,y)$ plane across the line $x=y$ .

The matrix $A=\mattwo {1}{K}{0}{1}$ is a shear. Describe the action of $A$ on the plane for different values of $K$ .

Determine a rotation matrix that maps the vectors $(3,4)$ and $(1,-2)$ onto the vectors $(-4,3)$ and $(2,1)$ respectively.

Find a $2\times 3$ matrix $P$ that projects three dimensional $xyz$ space onto the $xy$ plane. Hint: Such a matrix will satisfy $P\left (\begin {array}{c} 0 \\ 0 \\ z \end {array}\right ) = \vectwo {0}{0} \AND P\left (\begin {array}{c} x \\ y \\ 0 \end {array}\right ) = \vectwo {x}{y}.$

Show that every matrix of the form $\mattwo {a}{-b}{b}{a}$ corresponds to rotating the plane through the angle $\theta$ followed by a dilatation $cI_2$ where

$\begin{eqnarray*} c & = & \sqrt{a^2+b^2}\\ \cos\theta & = & \frac{a}{c} \\ \sin\theta & = & \frac{b}{c}. \end{eqnarray*}$

Using Exercise ?? observe that the matrix $\mattwo {3}{4}{-4}{3}$ rotates the plane counterclockwise through an angle $\theta$ and then dilates the planes by a factor of $c$ . Find $\theta$ and $c$ . Use map to verify your results.

In Exercises ?? – ?? use map to find vectors that are stretched and/or contracted to a multiple of themselves by the given linear mapping. Hint: Choose a vector in the MAP Display window and apply Map several times.

$A=\mattwoc {2}{0}{1.5}{0.5}$ .

$B=\mattwo {1.2}{-1.5}{-0.4}{1.2}$ .

$C=\mattwo {2}{-1.25}{0}{-0.5}$ .

In Exercises ?? – ?? use Exercise ?? and map to verify that the given matrices rotate the plane through an angle $\theta$ followed by a dilatation $cI_2$ . Find $\theta$ and $c$ in each case.

$A=\mattwo {1}{-2}{2}{1}$ .

$B=\mattwo {-2.4}{-0.2}{0.2}{-2.4}$ .

$C=\mattwoc {2.67}{1.3}{-1.3}{2.67}$ .

In Exercises ?? – ?? use map to help describe the planar motions of the associated linear mappings for the given $2\times 2$ matrix.

$A=\mattwo {\frac {\sqrt {3}}{2}}{\frac {1}{2}}{-\frac {1}{2}}{\frac {\sqrt {3}}{2}}$ .

$B = \mattwo {\frac {1}{2}}{-\frac {1}{2}}{\frac {1}{2}}{\frac {1}{2}}$ .

$C = \mattwo {0}{1}{1}{0}$ .

$D = \mattwo {1}{0}{0}{0}$ .

$E = \mattwo {\frac {1}{2}}{\frac {1}{2}}{\frac {1}{2}}{\frac {1}{2}}$ .

The matrix $A = \mattwo {0}{-1}{-1}{0}$ reflects the $xy$ -plane across the diagonal line $y=-x$ while the matrix $B=\mattwo {-1}{0}{0}{-1}$ rotates the plane through an angle of $180^\circ$ . Using the program map verify that both matrices map the vector $(1,1)$ to its negative $(-1,-1)$ . Now perform two experiments. First using the icon menu in map place a dog icon at about the point $(1,1)$ and move that dog by matrix $A$ . Then replace the dog in its orginal position near $(1,1)$ and move that dog using matrix $B$ . Describe the difference in the result.