Orthogonal Transformations - U-M LSA

1 Orthogonal TransformationsMath 217 Professor Karen Smith(c)2015 UM Math Deptlicensed under a Creative CommonsBy-NC-SA International :A linear transformationRnT Rnisorthogonalif|T(~x)|=|~x|for all~x :IfRnT Rnis Orthogonal , then~x ~y=T~x T~yfor all vectors~xand~ Last time you proved:1. An Orthogonal transformation is an The inverse of an Orthogonal transformation is also The composition of Orthogonal Transformations is with your table thegeometric intuitionof each of these statements. Why do they makesense, for example, SupposeT:R2 R2is given by left multiplication byA=[a cb d].AssumingTisorthogonal,illustrate an example of such aTby showing what it does to a unit square. Label your picturewitha,b,c,d. What do you think detAcan be?

2 Solution note:Your picture should show a square of sides 1 with vertices (0,0),(a,b),(c,d)and (a+c,b+d).The determinant ofAis either 1 or -1, since|detA|is the area ofthis :Ann nmatrix isorthogonalif its columns are IsA=[cos sin sin cos ] Orthogonal ? Is the map multiplication byA Orthogonal ? Why?Solution note:Yes, the columns are orthonormal since the (cos )2+ ( sin )2= 1,and (cos )( sin ) + (sin )(cos ) = Multiplication byAis rotation through anangle of (counterclockwise), so it preserves lengths and hence is IsB= 3/54/5 0 4/5 3/5 0001 Orthogonal ? IsBTorthogonal? Is the map multiplication byB Orthogonal ? Why? Describe it note:Yes, again, just check that the dot product of the columns is either 1or 0, depending on whether we dot a column with itself or a different column.

3 Sameis true forBT. Multiplication byAis it is a rotation Prove the followingTheorem:LetRnT Rnbe the linear transformationT(~x) =A~x, whereAis ann nmatrix. ThenTis Orthogonal if and only if the matrixAhas orthonormal columns.[Hint: Scaffold first. What are the two things you need to show?]Solution note:We need to show two things:1). IfTis Orthogonal , thenAhas orthonormal columns. AssumeTis columns ofAare [T(~e1)T(~e2).. T(~en)]. To check they are orthonormal, weneed two things:T(~ei) T(~ei) = 1 for alli, andT(~ei) T(~ej) = 0 ifi6=j. For thefirst, noteT(~ei) T(~ei) =||T(~ei)||2=||~ei||2= 1,with the first equality coming from the definition of the length of the vector and thesecond coming from the definition ofTbeing Orthogonal .

4 For the second,T(~ei) T(~ej) =~ei ~ejby the Theorem above. This is zero since the standard basis ). AssumeAhas orthonormal columns. We need to show that for any~x Rn,||T(~x)||=||~x||. Since the length is always a non-negative number, it suffices to show||T(~x)||2=||~x||2. That is, it suffices to showT(~x) T(~x) =~x ~x. For, this we take anarbitrary~xand write it in the basis{~e1,..,~en}. Note that~x ~x= (x1~e1+ +xn~en) (x1~e1+ +xn~en) = ij(xi~ei) (xj~ej) =x21+ + the third equality is using some basic properties of dot product (like foil ), andthe third equality is using the fact that the~eiare orthonormal so that (xi~ei) (xj~ej) = 0ifi6=j. On the other hand, we also haveT(~x) T(~x) =T(x1~e1+ +xn~en) T(x1~e1+ +xn~en)= ij(xiT(~ei)) (xjT(~ej))= ijxixjT(~ei) T(~ej) =x21+ +x2nwith the last equality coming from the fact that theT(~ei) s are the columns ofAandhence orthonormal.

5 WithAandBas in Problem C, computeATAandBTB. IsATorthogonal? do you notice?Solution note:The transposes of the Orthogonal matricesAandBare In general, it is true that the transpose of anothogonal matrix is Orthogonal AND that the inverse of an Orthogonal matrix is Prove that ifMis an Orthogonal matrix, thenM 1=MT. [Hint: writeMas a row of columnsandMTas a column of rows. ComputeMT M. ]Solution note:LetM= [~v1~ ~vn] where each~vi Rn. ThenMT= ~vT1~ ~vTn .So using block multiplication,MTMis the matrix whoseij=th entry is the product~vTi~vj, which is~vi ~ the~vi s are orthonormal, this is the identity Prove that the rows of an Orthogonal matrix are also orthonormal. [Hint: don t forget thatthe rows ofAare the columns ofAT.]

6 Remember that the inverse of an Orthogonal map is alsoorthogonal.]Solution note:SayAis Orthogonal . Then the mapTAis Orthogonal . Hence its inverseis Orthogonal , and so the matrix of the inverse, which isA 1is Orthogonal . By theprevious problem, we know also thatA 1=ATis Orthogonal . So since the columnsofATare orthonormal, which means the rows ofAare Prove the very importantTheoremon the first page saying that Orthogonal transformationspreserve dot products. Why does this tell us that Orthogonal Transformations preserve angles?[Hint: considerx+y.]Solution note:AssumeTis Orthogonal . So||T(x+y)||=||(x+y)||for allx,y Rn,by definition of Orthogonal . Hence also||T(x+y)||2=||(x+y)||2and soT(x+y) T(x+y) = (x+y) (x+y).Using linearity ofT, we have(T(x) +T(y)) (T(x) +T(y)) = (x+y) (x+y).

7 Now we expand both sides using basic properties of dot product:(T(x) +T(y)) (T(x) +T(y)) = (T(x) (T(x)) + 2T(x) T(y) + (T(y) T(y))=||T(x)||2+ 2T(x) T(y) +||T(y)|| this is equal to(x+y) (x+y) =x x+ 2x y+y y=||x||2+ 2x y+||y|| is,||T(x)||2+ 2T(x) T(y) +||T(y)||2=||x||2+ 2x y+||y|| Orthogonal ,||T(x)||=||x||and||T(y)||=||y ||,so we can cancel thesefrom both sides to get 2T(x) T(y) = 2x by 2, we see thatT(x) T(y) =x proof is Application that we are given a system ofnlinear equationsinnunknowns:A~x=~ AssumingAis invertible, express the solutions to this system in terms ofA 1and~ Assume thatA=QRis theQR-factorization ofA. What does this mean? What are thesizes of the matrices in this case?3. What happens if we multiply both sides of the equationA~x=~bbyQT?)

8 How might thissimplify the problem of finding the solution to this system?4. Suppose thatA= 44/5 11/50 1013 3/523/5 . Applying the Gram-Schmidt process to the columnsof this matrix we get 4/50 3/50 103/504/5 . Find the QR factorization note:We haveA=QRwhereQis the Orthogonal matrix above andRis 5110 10 1005 .5. Use your QR factorization to quickly solve the systemA~x= [0 0 25]Twithout rowreducing!.Solution note:To solveA~x=QR~x=~b,multiply both sides byQ 1,which isQTsinceQis Orthogonal . We have the equivalent systemR~x=QT~b= 15020 . This iseasy to solve becauseRis upper triangular. We getz= 4 from the bottom row. Thenthe second row gives 10y 4 = 0, soy= 2/5. The top row gives 5x+ 2/5 + 4 = 15,sox= 53 TRUE OR FALSE.

9 Justify. In all problems,Tdenotes a linear transformation fromRnto itself,andAis its matrix in the standard IfTis Orthogonal , thenx y=Tx Tyfor all IfTsends every pair of Orthogonal vectors to another pair of Orthogonal vectors, IfTis Orthogonal , thenTis An Orthogonal projection is IfAis the matrix of an Orthogonal transformationT, then the columns ofAare The transpose of an Orthogonal matrix is The product of two Orthogonal matrices (of the same size) is IfAis the matrix of an Orthogonal transformationT, thenAATis the identity IfA 1=AT, thenAis the matrix of an Orthogonal transformation ofRn.