Essay 2 on Matrices TRANSFORMATIONS via MATRICES
James Tanton
It follows that REVIEW OF KNOWN RESULTS AOOAA+=+= Matrices arise in a variety of scenarios. We first motivated them by encoding network diagrams AI⋅=⋅ I A = A with matrices. From this a natural arithmetic for matrices arose—their scalar multiplication, Also, the determinant of A is ||A= ad − bc addition and multiplication. One can extend this (also often denoted det ( A)), and we have arithmetic to look at matrix subtraction and the possible existence of matrix inverses. A−1 exists precisely when |A |0≠ . We also saw in the previous essay that we have a complete theory of arithmetic for the 22× −1 1 db− square matrices. In quick summary: In this case A = and we have − ||A ca ab ef If A = and B = , and A−−11⋅=⋅ A AA = I. cd gh 00 In this essay we shall focus on 22× matrices O = is the zero matrix, and 00 and look at geometric interpretations of them.
10 I = is the identity matrix 01 TRANSFORMING POINTS OF THE PLANE we set = ka kb A point in the plane P( xy, ) can be kA = regarded as a matrix with one row and two kc kd columns. As such we can apply the arithmetic of matrices on it. If A is a 22× matrix, then, aeb++ f since P is a 12× matrix, we can compute the AB+= cgdh++ product PA and get a new 12× matrix, that is, a new point in the plane. In this way, a matrix ae++ bg af bh A can be viewed as an “operation” that moves AB = points of the plane around. ce++ dg cf dh (multiply the entries of row i of A by the Annoying Convention: Notice that the product matching entries of column j of B and add.) AP cannot be computed—the dimensions of the matrices here don’t appropriately match. But, sadly, society has settled on the notation Fx( ) for transforming a number x into some
new number by a procedure being call F 10 (“function” notation). Since our 22× matrix A One checks that the matrix H = has 01− is being used to transform points, we would prefer to use notation for this that appears as the same effect on points in the plane.
AP , with the name of the operator on the left. 10 xx = One simple way to have this matrix product 01−− yy make sense is to write points as 21× matrices, that is, to write the coordinates of a point P in Thus a horizontal reflection is represented by x the 22× matrix H . a column: P = . In this way each 22× y Performing the transformation H twice (the matrix A gives a transformation of points in reflection twice) clearly returns points back to the plane given as follows. 2 their original positions. Thus H maps a point xx x 2 A . back to itself. We have then that H is yy y the identity transformation, giving
2 EXAMPLES OF TRANSFORMATIONS HI= .
The Identity Transformation (One can check this is true by computing the matrix product directly. Do it!) Also, since the 10 The identity matrix I = does not alter reflection “undoes” itself, we must have 01 HH−1 = . (One can check this is true too by points in the plane. One checks that computing the matrix inverse to H .)
10 xx A reflection about the vertical axis takes a point = . 01 yy x −x in the plane to the point . y y Three Basic Reflections A reflection about the horizontal axis takes a x x point in the plane to the point . y −y
One can check it can be represented by the −10 22× matrix V = . It is also clear 01 from the geometry that 2 = and −1 = . VI VV
A reflection about the northeast diagonal line x As such it takes a point and maps it to x y y takes a point in the plane to the point . y x kx . One checks that the 22× matrix ky k 0 Ek = has the same effect on points. 0 k
k0 x kx = 0 k y ky
1 One can check it can be represented by the A dilation/enlargement by a factor undoes a 01 k 22× matrix D = . It is clear from the dilation/enlargement by a factor k so we 10 expect EE−1 = . One can check that this is 2 = −1 = k 1 geometry that DI and DD. k
true by computing the matrix inverse of E . Challenge 1: Find the 22× matrix that k corresponds to a reflection across the southeast diagonal line. Challenge 2: What if we allow k to be zero or negative? For example, can you identify the Dilations/Enlargements geometric effect of the matrix E0 and of E−1 ? A dilation (also sometimes known as an enlargement) takes each point of the plane and A Basic Rotation pushes it radially away from the origin by some A counterclockwise rotation of 900 about the fixed positive factor k . (If k is between 0 and x −y 1, then the dilation is pushing points closer to origin takes a point to the point . the origin and the name “enlargement” is a y x misnomer.)
One checks that this is represented by 01− R = . The inverse rotation, a 90 10 clockwise rotation about the origin, is given by
−1 01 R = . −10
−10 Example: Consider the matrix . SOMETHING WORTH NOTING 01−
1 −1 Perhaps try this challenge, or feel free to simply We see now that it takes to , and read through it and then read on. 0 0 0 0 Challenge 3: Is there a transformation of the to , and so rotates each of these 1 −1 plane given by a 22× matrix that takes the 1 2 0 4 points 180 about the origin. (Draw the points point to and the point to ? on a graph.) It also keeps the origin fixed in 0 −3 1 9 place. This matrix thus does indeed represent a If so, what is the matrix? 180 rotation about the origin. (One can swiftly
also see that the two 90 rotations discussed For the matrix A in the opening section, one earlier are correctly representations.) checks that
Example: In Challenge 1, the transformation ab 1 a = , the first column of A 1 0 cd 0 c described there takes to , and 0 −1 ab 0 b = , the second column of A . 0 −1 cd 1 d to (draw it!) and it fixes the origin. 1 0
So given a matrix A we can read off what it 01− The matrix does the same. does to two basic points of the plane by looking −10 at each of its two columns. Example: The answer to challenge 2 is the 24 matrix . −39
Challenge 4: Consider the geometric transformation that shifts all points in the plane three units to the right. Explain why this geometric transformation cannot be represented by a matrix.
Challenge 5: Is there a transformation of the Also, one sees that plane given by a 22× matrix that takes the 2 1 4 0 point to and the point to ? ab 00 −3 0 9 1 = cd 00 If so, what is the matrix?
and so the origin stays fixed in place. These observations often let you swiftly ascertain what the geometric effect of given matrix might be.
Practice 6: Consider the following diagram. DETERMINANTS One can measure the “strength” of transformation by examining what it does to a fundamental region of the plane. It seems reasonable to regard the unit square with 0 1 0 1 points , ,, and as vertices as 0 0 1 1 the fundamental unit of area.
a) Draw the image of figure A −1 i) under a translation given by . If a transformation is given by 22× matrix −3 ab ii) under the transformation represent by the A = , then it transforms each of these cd 10 matris . four vertices to the following points. 01− ab 00 b) Name a single transformation that maps = figure A to figure B. cd 00 ab 1 a c) Write a matrix that represents a = cd 0 c transformation that takes figure A to figure C. ab 0 b = cd 1 d ab 1 ab+ = cd 1 cd+
These are the vertices of a parallelogram.
The parallelogram shown sits inside a rectangle of area (acbd++)( ) . And subtracting off the areas of the smaller triangles and rectangles you see, one can show that the area of parallelogram is ad− bc , which is the determinant of the matrix A !
Comment: This picture assumes that the parallelogram sits fully within the first quadrant of the plane. One can check that no matter how the parallelogram is situated—one can verify all possible cases by hand—that the area of the parallelogram is ||ad− bc , the absolute value of the determinant of the matrix.
The determinant of a matrix gives the factor by which areas change under the geometric effect induced by that matrix.
Examples: Rotations do not change the areas of objects and, indeed, the determinant of each of the rotation matrices we discussed is 1. The absolute value of each reflection matrix is 1 and, indeed, reflections preserve areas. The
determinant of a dilation/enlargement Ek is k 2 and indeed all areas change by the factor k 2 under enlargement. Any matrix transformation that collapses a region of positive area to zero area cannot be inverted.
Challenge 5: Is there a transformation of the PROBLEM SOLUTIONS plane given by a 22× matrix that takes the 2 1 4 0 Find the 22× matrix that point to and the point to ? Challenge 1: −3 0 9 1 corresponds to a reflection across the southeast diagonal line. If so, what is the matrix?
Solution: The solution is given in the essay! Solution: Yes!
Challenge 2: What if we allow k to be zero or 24 1 2 The matrix M = takes to negative? For example, can you identify the −39 0 −3 geometric effect of the matrix E0 and of E−1 ? 0 4 −1 and the point to . The matrix M 1 9 Solution: E0 is the zero matrix and its (which exists since det(M ) =+= 18 12 30 is geometric effect is to map all points to the non-zero) does the reverse of this, which is origin. The matrix E− maps each point P to a 1 what the question called for. matching point Q on the other side of the origin O so that O is the midpoint of PQ . Practice 6: Consider the following diagram. (This is the same as a 180 rotation about the origin.)
Challenge 3: Is there a transformation of the plane given by a 22× matrix that takes the 1 2 0 4 point to and the point to ? 0 −3 1 9 If so, what is the matrix?
Solution: This is answered in the essay. a) Draw the image of figure A −1 i) under a translation given by . Challenge 4: Consider the geometric −3 transformation that shifts all points in the plane ii) under the transformation represent by the three units to the right. Explain why this 10 geometric transformation cannot be matrix . represented by a matrix. 01−
Solution: Every transformation represented by b) Name a single transformation that maps a matrix maps the origin to the origin: figure A to figure B. ab 00 = . c) Write a matrix that represents a cd 00 transformation that takes figure A to figure C. Since a translation does not map the origin to the origin, no translation is represented by a Solution: matrix! i) It’s the green square shown. ii) It’s the pink square shown. (Observe what this matrix does to the two fundamental points
1 0 and to see it matches a reflection 0 1 about the horizontal line.)
b) A dilation or enlargement with scale factor 1 − . 2
c) Although a translation takes A to C, a translation cannot be represented by a matrix. Instead a reflection about the diagonal line 1 yx= takes A to C. Such a reflection takes 0 0 0 1 to and to and so is given by the 1 1 0 01 matrix . 10