Spring, 2008 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 1, Corrected Version

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1 Spring, 008 CIS 610 Adanced Geometric Methods in Compter Science Jean Gallier Homework 1, Corrected Version Febrary 18, 008; De March 5, 008 A problems are for practice only, and shold not be trned in. Problem A1. a Find two symmetric matrices, A and B, sch that AB is not symmetric. b Find two matrices, A and B, sch that e A e B e A+B. Try A = π and B = π Problem A. a If K = R or K = C, recall that the projectie space, PK n+1, is the set of eqialence classes of the eqialence relation,, onk n+1 {0}, defined so that, for all, K n+1 {0}, iff = λ, for some λ K {0}. The map, p:k n+1 {0} PK n+1, is the projection mapping any nonzero ector in K n+1 to its eqialence class modlo. WeletRP n = PR n+1 andcp n = PC n+1. Proe that for any n 0, there is a bijection between PK n+1 andk n PK n which allows s to identify them. b Proe that RP n and CP n are connected and compact. Hint. If S n = {x 1,...,x n+1 K n+1 x x n+1 =1}, proe that ps n =PK n+1, and recall that S n is compact for all n 0 and connected for n 1. For n =0,PK consists of a single point. Problem A3. Recall that R and C can be identified sing the bijection x, y x+iy. Also recall that the sbset U1 C consisting of all complex nmbers of the form cos θ + i sin θ is homeomorphic to the circle S 1 = {x, y R x + y =1}. If c: U1 U1 is the map defined sch that cz =z, 1

2 proe that cz 1 =cz iffeitherz = z 1 or z = z 1, and ths that c indces a bijectie map ĉ: RP 1 S 1. Proe that ĉ is a homeomorphism remember that RP 1 is compact. B problems mst be trned in. Problem B1 0 pts. Let A =a ij be a real or complex n n matrix. 1 If λ is an eigenale of A, proe that there is some eigenector = 1,..., n ofa for λ sch that max i =1. 1 i n If = 1,..., n is an eigenector of A for λ as in 1, assming that i, 1 i n, is an index sch that i = 1, proe that λ a ii i = n a ij j, j=1 j i and ths that λ a ii n a ij. Conclde that the eigenales of A are inside the nion of the closed disks D i defined sch that { n } D i = z C z a ii a ij. j=1 j i j=1 j i Remark: This reslt is known as Gershgorin s theorem. Problem B 10. Recall that a real n n symmetric matrix, A, ispositie semi-definite iff its eigenales, λ 1,...,λ n are non-negatie i.e., λ i 0 for i =1,...,nandpositie definite iff its eigenales are positie i.e., λ i > 0 for i =1,...,n. a Proe that a symmetric matrix, A, is positie semi-definite iff X AX 0, for all X 0X R n and positie definite iff X AX > 0, for all X 0X R n. b Proe that for any two positie definite matrices, A, B, for all λ, µ R, withλ, µ 0 and λ + µ>0, the matrix λa + µb is still symmetric, positie definite. Dedce that the set of n n symmetric positie definite matrices is conex in fact, a cone. Problem B3 40 pts. a Gien a rotation matrix cos θ sin θ R = sin θ cos θ,

3 where 0 <θ<π, proe that there is a skew symmetric matrix B sch that R =I BI + B 1. b If B is a skew symmetric n n matrix, proe that λi n B and λi n + B are inertible for all λ 0, and that they commte. c Proe that R =λi n BλI n + B 1 is a rotation matrix that does not admit 1 as an eigenale. Recall, a rotation is an orthogonal matrix R with positie determinant, i.e., detr = 1. d Gien any rotation matrix R that does not admit 1 as an eigenale, proe that there is a skew symmetric matrix B sch that R =I n BI n + B 1 =I n + B 1 I n B. This is known as the Cayley representation of rotations Cayley, e Gien any rotation matrix R, proe that there is a skew symmetric matrix B sch that R = I n BI n + B 1. Problem B4 60. a Consider the map H: R 3 R 4 defined sch that x, y, z xy, yz, xz, x y. Proe that when it is restricted to the sphere S in R 3, we hae Hx, y, z =Hx,y,z iff x,y,z =x, y, z orx,y,z = x, y, z. In other words, the inerse image of eery point in HS consists of two antipodal points. Proe that the map H indces an injectie map from the projectie plane onto HS, and that it is a homeomorphism. b The map H allows s to realize concretely the projectie plane in R 4 as an embedded manifold. Consider the three maps from R to R 4 gien by ψ 1, = ψ, = ψ 3, = + +1, + +1, + +1, + +1, + +1, + +1, + +1,, , 1, , Obsere that ψ 1 is the composition H α 1,whereα 1 : R S is gien by, + +1, + +1, 1,

4 that ψ is the composition H α,whereα : R S is gien by, + +1, , + +1 and ψ 3 is the composition H α 3,whereα 3 : R S is gien by 1, + +1, + +1, + +1 Proe that each ψ i is injectie, continos and nonsinglar i.e., the Jacobian is neer zero. Proe that if ψ 1, =x, y, z, t, then., y + z 1 4 and y + z = 1 4 iff + =1. Proe that and are soltions of the qadratic eqations y + z z + z = 0 y + z y + y = 0. Proe that if y + z 0,then = z1 1 4y + z y + z if + 1, else = z y + z if + 1, y + z and there are similar formlae for. Proe that the expression giing in terms of y and z is continos eerywhere in {y, z y + z 1 } and similarly for the expression giing 4 in terms of y and z. Conclde that ψ 1 : R ψ 1 R is a homeomorphism onto its image. Therefore, U 1 = ψ 1 R is an open sbset of HS. Remark: From the eqations aboe, yo can proe that + +1 is a root of the eqation y + z D D +1=0. Then, else D = 1 1 4y + z y + z D = y + z y + z 4 if + 1, if + 1.

5 Proe that if ψ, =x, y, z, t, then and are soltions of qadratic eqations with coefficients inoling x and y; find explicit formlae as for ψ1 1 and conclde that ψ : R ψ 3 R is a homeomorphism onto its image. The set U = ψ R isanopen sbset of HS. Proe that if ψ 3, =x, y, z, t, then and are soltions of qadratic eqations with coefficients inoling x and z. As for ψ 1, conclde that ψ 3 : R ψ 3 R is a homeomorphism onto its image. The set U 3 = ψ 3 R is an open sbset of HS. Proe that the nion of the U i s coers HS. Conclde that ψ 1,ψ,ψ 3 are parametrizations of RP as a manifold in R 4. Proe that if x, y, z, t HS, then x y + x z + y z = xyz xz y = yzt. The zero locs of these eqations strictly contains HS, proe it. This is a famos mistake of Hilbert and Cohn-Vossen in Geometry and the Immagination! In an attempt to fix this bg, proe that when yo express x in terms of y and z sing ψ 1, yo get the eqation x y + x z + y z = xyz. When yo express t in terms of y and z sing ψ 1, yo get the eqation y + z z y + t =tz y. When yo express t in terms of x and y sing ψ, yo get the eqation 4x + y x + y t +x + y =x + y. When yo express t in terms of x and z sing ψ 3, yo get an eqation similar to the preios one. Do these for eqations define exactly HS? I sspect they do! c Inestigate the srfaces in R 3 obtained by dropping one of the for coordinates. Show that there are only two of them the Steiner Roman srface and the crosscap, p to a rigid motion. Problem B5 40. a Consider the map, f: GL + n Sn, gien by fa =A A I. Check that df AH =A H + H A, for any matrix, H. b Consider the map, f: GLn R, gienby fa =deta. 5

6 Proe that df IB = trb, the trace of B, for any matrix B here, I is the identity matrix. Then, proe that df AB =detatra 1 B, where A GLn. c Use the map A deta 1toproethatSLn is a manifold of dimension n 1. d Let J be the n +1 n + 1 diagonal matrix In 0 J =. 0 1 We denote by SOn, 1 the grop of real n +1 n + 1 matrices SOn, 1 = {A GLn +1 A JA = J and deta =1}. Check that SOn, 1 is indeed a grop with the inerse of A gien by A 1 = JA J this is the special Lorentz grop. Consider the fnction f: GL + n +1 Sn +1,gienby fa =A JA J, where Sn + 1 denotes the space of n +1 n + 1 symmetric matrices. Proe that df AH =A JH + H JA for any matrix, H. Proe that df A is srjectie for all A SOn, 1 and that SOn, 1 is a manifold of dimension nn+1. Problem B6 0 pts. a Gien any matrix a b B = sl, C, c a if ω = a + bc and ω is any of the two complex roots of a + bc, proe that if ω 0,then e B =coshωi+ sinh ω ω and e B = I + B, ifa + bc = 0. Obsere that tre B =coshω. Proe that the exponential map, exp: sl, C SL, C, is not srjectie. For instance, proe that is not the exponential of any matrix in sl, C. B, 6

7 Problem B7 50 pts. Recall that for any matrix A = 0 c b c 0 a, b a 0 if we let θ = a + b + c and B = a ab ac ab b bc, ac bc c then the exponential map, exp: so3 SO3, is gien by exp A = e A =cosθi 3 + sin θ 1 cos θ A + B, θ θ or, eqialently, by e A = I 3 + sin θ 1 cos θ A + A, θ θ if θ kπ k Z, with exp0 3 =I 3 Rodriges s formla a Let R SO3 and assme that R I and trr 1. Then, proe that a log of R i.e., a skew symmetric matrix, S, sothate S = R isgienby logr = θ sinθ R RT, where 1 + cos θ = trr and0<θ<π. b Now, assme that trr = 1. In this case, show that R is a rotation of angle π, that R is symmetric and has eigenales, 1, 1, 1. Assming that e A = R, Rodriges formla becomes R = I + π A, so A = π R I. If we let S = A/π, we see that we need to find a skew-symmetric matrix, S, sothat S = 1 R I =C. Obsere that C is also symmetric and has eigenales, 1, 1, 0. Ths, we can diagonalize C, as C = P P,

8 and if we let S = P P, check that S = C. c From a and b, we know that we can compte explicity a log of a rotation matrix, althogh when θ 0, we hae to be carefl in compting sin θ ; in this case, we may want to θ se sin θ =1 θ θ 3! + θ4 5! +. Gien two rotations, R 1,R SO3, there are three natral interpolation formlae: e 1 tlogr 1+t log R ; R 1 e t logr 1 R ; e t logr R 1 R 1, with 0 t 1. Write a compter program to inestigate the difference between these interpolation formlae. The position of a rigid body spinning arond its center of graity is determined by a rotation matrix, R SO3. If R 1 denotes the initial position and R the final position of this rigid body, by compting interpolants of R 1 and R,wegetamotionoftherigidbody and we can create an animation of this motion by displaying seeral interpolants. The rigid body can be a fnny object, for example a banana, a bottle, etc. TOTAL: 40 points. 8