MATRIX TRANSFORMATIONS
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1 CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B from A to B assigns to each element a of A a (unique) element f(a) of B. The set A on which the function is defined is referred to as the domain of the function f : A B. The set B into which the domain is mapped b f is referred to as the codomain of the function f. A B a f(a) f : A B For man common functions the domain and codomain are the sets of real numbers. concerned with functions for which the domain and codomain are vectors. We are
2 CHAPTER 5. MATRIX TRANSFORMATIONS Definition Let A and B be vectors and if f is a function with domain A and codomain B, then we sa that f is a transformation from A to B or that maps A to B, which we denote b writing f : A B In the special case where A = B, the transformation is also called and operator on V. We seek a transformation that maps the column vector in R n into the column vector in R n b multipling b the matri A. We call this a matri transformation and we denote it b T A : R n R n R n R n T A () T A : R n R n In shorthand, we have T A () = [T ] The matri transformation T A is called multiplication b A, and the matri A is called the standard matri for the transformation. The following theorem lists the four basic properties of matri transformations that follow from properties of matri multiplication.
3 3 CHAPTER 5. MATRIX TRANSFORMATIONS 3 Theorem For ever matri A, the matri transformation T A : R n R m has the following properties for all vectors u and v in R n and for ever scalar k: i T A () = ii T A (k u) = kt A ( u) iii T A ( u + v) = T A ( u) + T A ( v) iv T A ( u v) = T A ( u) T A ( v) The geometric effect of a matri transformation T A : R n R m is to map each vector (point) in R n into a vector (point) in R m. Remark There is a wa of finding the standard matri A for a matri transformation from R n to R m b considering the effect of that transformation on the standard basis vectors for R n. The standard basis vectors for R are e = (, ) and e = (, ). The standard basis vectors for R 3 are e = (,, ), e = (,, ) and e 3 = (,, ). To eplain the idea, in general, suppose that the standard matri A is unknown and that e, e,..., e n are the standard basis vectors for R n. transformation T A are Suppose also that the images of these vectors under the T A (e ) = Ae, T A (e ) = Ae,..., T A (e n ) = Ae n We can write that A = [ ] T A (e ) T A (e )... T A (e n ) In summar, we have the following procedure for finding the standard matri for a matri transformation:
4 4 CHAPTER 5. MATRIX TRANSFORMATIONS 4 Step : Find the images of the standard basis vectors e, e,..., e n for R n. in column form. Step : Construct the matri that has the images obtained in Step as its successive columns. This matri is the standard matri for the transformation.. Reflection Operators Some of the most basic matri operators on R 3 are those that map each point into its smmetric image about a fied plane. These are called reflection operators. The following diagrams show the standard matrices for the reflections about the coordinate planes in R 3. In each case the standard matri was obtained b finding the images of the standard basis vectors, converting those images to column vectors, and then using those column vectors as successive column of the standard matri. The following matrices will perform orthogonal reflections ABOUT the plane, the z plane and the z plane respectivel. Reflection about the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T =
5 5 CHAPTER 5. MATRIX TRANSFORMATIONS 5 Reflection about the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T = Reflection about the plane: T (,, z) = (,, z) z T () (,, z) (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T =
6 6 CHAPTER 5. MATRIX TRANSFORMATIONS 6. Projection Operators Matri operators on R 3 that map each point into its orthogonal projection on a fied plane are called projection operators (or more precisel, orthogonal projection operators). The following are the standard matrices for the orthogonal projections on the coordinate aes in R 3. Again, in each case the standard matri was obtained b finding the images of the standard basis vectors, converting those images to column vectors, and then using those column vectors as successive column of the standard matri. Orthogonal Projection onto the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T = The following matrices will perform orthogonal projection ONTO the plane, the z plane and the z plane respectivel. A = A = A =
7 7 CHAPTER 5. MATRIX TRANSFORMATIONS 7 Eample To determine the coordinates of the point (,, 3) after a reflection in the z plane, we have T A () = [T ] =. 3 = 3 i.e., (,, 3) (,, 3)
8 8 CHAPTER 5. MATRIX TRANSFORMATIONS 8.3 Rotation Operators Matri Operators on on R and R 3 that move points along circular arcs are called rotation operators. Let us consider how to find the standard matri for the rotation operator T : R R that moves points anti-clockwise about the origin through an angle θ. e ( sin θ, cos θ) (cos θ, sin θ) θ θ e The images of the standard basis vectors in R are T (e ) = T (, ) = (cos θ, sin θ) T (e ) = T (, ) = ( sin θ, cos θ) Hence, the standard matri for this transformation is ( cos θ sin θ A = sin θ cos θ ) In keeping with common notation, we represent this operator b R θ, i.e., ( ) cos θ sin θ R θ = sin θ cos θ
9 9 CHAPTER 5. MATRIX TRANSFORMATIONS 9 (w, w ) w (, ) θ Note In the plane, anti-clockwise angles are positive and clockwise angles are negative. The rotation matri for a clockwise rotation of θ radians can be obtained b replacing θ b θ. Note that cos( θ) = cos θ and sin( θ) = sin θ. After simplification this ield ( cos θ sin θ R θ = sin θ cos θ ) A rotational operator in R 3 is a matri operator that rotates each vector in R 3 about some rotation ais through fied angle θ. A rotation of vectors in R 3 is usuall described in relation to a ra emanating from the origin, called the ais of rotation. As a vector revolves around the ais of rotation, it sweeps out some portion of a cone. The angle of rotation, which is measured in the base of the cone, is described as clockwise or anti-clockwise in relation to a viewpoint that is along the ais of rotation looking towards the origin. z (w, w, w 3 ) w θ (,, z) Ais of Rotation
10 CHAPTER 5. MATRIX TRANSFORMATIONS The following three matrices will rotate an vector anti-clockwise about the, and z aes through an angle of θ respectivel. For each of these rotations one of the components is unchanged and the relationship between the other components can be derived b the same procedure used to derive rotational matrices in R. About the -ais, -ais and z-ais respectivel, we have R θ = R θ = R z θ = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ The following three matrices will rotate an vector clockwise about the, and z aes through an angle of θ respectivel. About the -ais, -ais and z-ais respectivel, we have R θ = R θ = R z θ = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ
11 CHAPTER 5. MATRIX TRANSFORMATIONS Eample Using a matri transformation rotate the triangle ABC whose vertices are the points A(,, ), B(,, ), C(,, ) through 6 o anti-clockwise about the z-ais as follows: A β B α γ C of 6 o Firstl, the following matri will rotate an vector anti-clockwise about the z-aes through an angle Now R 6 o = R 6 o = [T ] = R 6 o = [T ] = R 6 o = [T ] = cos 6 o sin 6 o sin 6 o cos 6 o = = = = Finall, we have A ( 366, 366, ), B ( 34, 3, ), C ( 366, 366, ).
12 CHAPTER 5. MATRIX TRANSFORMATIONS Remark Consider the matri A = This matri represents a clockwise rotation about the z-ais through 9 o. Notice that the inverse matri is A = which is an anti-clockwise rotation about the z-ais through 9 o. It is eas to see here that the inverse matri is simpl the transpose of the original matri A. This is ver common in computer graphics and an matri with this propert is called an orthogonal matri. Definition A square matri A is an orthogonal matri if A = A t. This can alwas be tested b multipling A.A t and see if the result is the identit matri. Notice the the three clockwise rotational matrices R θ,r θ and R z θ are each orthogonal since multipling each matri b its transpose will ield I the identit matri. So, for eample, R θ.(r θ) t = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ = This can be confirmed for the matri R θ and R z θ in a similar wa. When an orthogonal matri is used to rotate vectors, it will keep the lengths of the vectors preserved as will the angle between the vectors. So an orthogonal matri ma represent a rotation.
13 3 CHAPTER 5. MATRIX TRANSFORMATIONS 3 Eample Consider the matri A = To show that A is orthogonal, note that a square matri A is an orthogonal matri if A = A t. Pre-multipling both sides b A ield A.A = I = A.A t Now A.A t = =.4 Some Eercises Eercise Using a matri transformation rotate the triangle ABC whose vertices are the points A(3, 4, 8), B(, 3, 5), C(, 6, 7) through 3 o anti-clockwise about the -ais. Eercise Using a matri transformation rotate the cube whose corners are the points A(, 5, ), B(4, 5, ), C(,, ), D(4,, ) E(, 5, 4), F (4, 5, 4), G(,, 4), H(4,, 4) through 45 o anti-clockwise about the -ais.
14 4 CHAPTER 5. MATRIX TRANSFORMATIONS 4.5 Dilations and Contractions If k is a non-negative scalar, then the operator T () = k in R or R 3 has the effect of increasing or decreasing the length of each vector b a factor of k. If k the operator is called a contraction with factor k, and if k > it is called a dilution with factor k. If k =, then T is the identit operator and can be regarded as either a contraction or a dilation. (, ) (, k) k > (, ) (k, ) k (, ) (, k) (, ) (k, ) The following matrices will perform a contraction or a dilation with factor k in R. A = ( k k ) The following matrices will perform a contraction or a dilation with factor k in R 3.
15 5 CHAPTER 5. MATRIX TRANSFORMATIONS 5 A = k k k In a dilation or contraction in R or R 3, all coordinates are multiplied b a factor k. If onl one of the coordinates is multiplied b b a factor k, then the resulting operator is called an epansion or compression with factor k. Consider an epansion or compression with factor k in the -direction. k > (, ) (, ) (, ) (k, ) k (, ) (, ) (, ) (k, ) The following matrices will perform a epansion or compression with factor k in R in the -direction. A = ( k )
16 6 CHAPTER 5. MATRIX TRANSFORMATIONS 6 Consider an epansion or compression with factor k in the -direction. (, ) (, k) k > (, ) (, ) k (, ) (, k) (, ) (, ) The following matrices will perform a epansion or compression with factor k in R in the -direction. A = ( k ) The following matrices will perform a epansion or compression with factor k in R 3 in the -direction, in the -direction and in the z-direction respectivel. A = A = A = k k k
17 7 CHAPTER 5. MATRIX TRANSFORMATIONS 7.6 Properties of Matri Transformations We seek to show that if several matri transformations are performed in succession, then the same result can be obtained b a single matri transformation that is chosen appropriatel. Firstl, we recall the definition of a composite function. Let A, B and C be sets, let f : A B be a function A to B, let g : B C be a function from B to C. Then there is a function g f : A C obtained b composing the functions f and g. This function is defined at each element a of A b the formula (g f)(a) = g(f(a)) In other words, in order to appl the composition function g f to an element a of A, we first appl the function f to the element a, and then we appl the function g to the resulting element f(a) of B to obtain an element g(f(a)) of C. A B C f g a f(a) g(f(a)) gof Remark Note that g f denotes the composition function f followed b g. The functions are specified in this order (which ma at first seem odd) in order that (g f)(a) = g(f(a)) for all elements a of the domain A of the function f. Eample Let f : R R defined b f() = and let g : R R defined b g() = + 5. Now (g f)() = g(f()) = g( ) = + 5. Also (f g)() = f(g()) = f( + 5) = Eercise Let f : R R defined b f() = ( + ) and let g : R R defined b g() = sin. Determine the composite functions g f and f g.
18 8 CHAPTER 5. MATRIX TRANSFORMATIONS 8 Now, consider the following matri transformations T A : R n R n T B : R n R n The composition of T A with T B will be denoted b T B T A Note that T B T A denotes the composition transformation T A followed b T B, i.e., the transformation T A is performed first. This composition of transformations is defined as (T B T A )() = [T B ][T A ] R n R n R n T T T () T (T ()) T T Eample Let T A : R R such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R R that first reflects about the line =, then rotates through an angle of 8 o anti-clockwise about the origin, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T T
19 9 CHAPTER 5. MATRIX TRANSFORMATIONS 9 where T is the reflection about the line = and T rotates through an angle of 8 o anti-clockwise about the origin. The standard matrices for these operations are [T ] = [T ] = ( ) ( ) thus, it follows, that the standard matri A = T T is [T ][T ] = ( ) (. ) = ( ) Therefore A = ( ) Eample Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R 3 R 3 that first rotates a vector anti-clockwise about the z-ais through an angle θ, then reflects the resulting vector about the z-plane, and then projects that vector orthogonall onto the -plane, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T 3 T T where T is the rotation about the z-ais, T is the reflection about the z-plane, and T 3 is the orthogonal projection n the -plane. The standard matrices for these operations are [T ] = [T ] = cos θ sin θ sin θ cos θ [T 3 ] =
20 CHAPTER 5. MATRIX TRANSFORMATIONS thus, it follows, that the standard matri A = T 3 T T is Therefore [T 3 ][T ][T ] = A =.. cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ Eample Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R 3 R 3 that first rotates a vector anti-clockwise about the -ais through an angle 3 o, then rotates the resulting vector anti-clockwise about the z-ais through an angle of 3 o, and then contracts that vector b a factor of k = 4, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T 3 T T where T is the rotation anti-clockwise about the -ais through an angle 3 o, T is the rotation anti-clockwise about the z-ais through an angle 3 o, and T 3 contraction b a factor of k = 4. The standard matrices for these operations are [T ] = [T ] = 3 4 [T 3 ] = 4 4 thus, it follows, that the standard matri A = T 3 T T is
21 CHAPTER 5. MATRIX TRANSFORMATIONS [T 3 ][T ][T ] = Therefore A = Eercise Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. Find the standard matri for the stated composition in R 3. i A reflection about the -plane, followed b a reflection about the z-plane, followed b an orthogonal projection on the z-plane. ii A rotation of 7 o anti-clockwise about the -ais, followed b a rotation of 9 o anti-clockwise about the -ais, followed b a rotation of 8 o anti-clockwise about the z-ais.
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