Solutions hapter 8 sds 8 Simple Dynamical Systems 8 Solve the following initial value problems: (a) du du = u u() = (b) = u u() = dt dt du (c) dt (a) u(t) = e t (b) u(t) = e (t ) (c) u(t) = e (t+) = u u( ) = 8 Suppose a radioactive material has a half-life of years What is the decay rate γ? Starting with an initial sample of grams how much will be left after years? years? years? γ = log / 69 fter years: 9 gram; after years gram; after years 977 gram 8 arbon-4 has a half-life of 7 years Human skeletal fragments discovered in a cave are analyzed and found to have only 64% of the carbon-4 that living tissue would have How old are the remains? Solve e (log )t/7 = 64 for t = 7 log 64/ log = 9 years 84 Prove that if t is the half-life of a radioactive material then u(nt ) = n u() Explain the meaning of this equation in your own words (log )t/t y (86) u(t) = u() e = u() t/t = n u() when t = nt fter every time period of duration t the amount of material is cut in half 8 bacteria colony grows according to the equation du/dt = u How long until the colony doubles? quadruples? If the initial population is how long until the population reaches million? u(t) = u() e t To double we need e t = so t = log / = To quadruple takes twice as long t = 664 To reach million needs t = log 6 / = 67 86 Deer in Northern Minnesota reproduce according to the linear differential equation du dt = 7u where t is measured in years If the initial population is u() = and the environment can sustain at most deer how long until the deer run out of resources? The solution is u(t) = u() e 7 t For the given initial conditions u(t) = when t = log(/)/7 = 964 years 87 onsider the inhomogeneous differential equation du = au + b where a b are constants dt (a) Show that u = b/a is a constant equilibrium solution (b) Solve the differential equation Hint: Look at the differential equation satisfied by v = u u (c) Discuss the stability of the equilibrium solution u evv 9/9/4 4 c 4 Peter J Olver
(a) If u(t) u = b du then = = au + b hence it is a solution a dt (b) v = u u satisfies dv dt = av so v(t) = cea t and so u(t) = ce a t b a (c) The equilibrium solution is asymptotically stable if and only if a < and is stable if a = 88 Use the method of Exercise 87 to solve the following initial value problems: (a) du du du = u u() = (b) = u + u() = (c) = u + 6 u() = dt dt dt (a) u(t) = + e t (b) u(t) = (c) u(t) = e (t ) 89 The radioactive waste from a nuclear reactor has a half-life of years Waste is continually produced at the rate of tons per year and stored in a dump site (a) Set up an inhomogeneous differential equation of the form in Exercise 87 to model the amount of radioactive waste (b) Determine whether the amount of radioactive material at the dump increases indefinitely decreases to zero or eventually stabilizes at some fixed amount (c) Starting with a brand new site how long until the dump contain tons of radioactive material? (a) du dt = log u = / log 7 tons (c) The solution is u(t) = when t = log log log u + 69 u + (b) Stabilizes at the equilibrium solution 4years log h exp log t i which equals 8 Suppose that hunters are allowed to shoot a fixed number of the Northern Minnesota deer in Exercise 86 each year (a) Explain why the population model takes the form du = 7u b where b is the number killed yearly (Ignore the season aspects of hunting) (b) If b = how long until the deer run out of resources? Hint: See Exercise dt 87 (c) What is the maximal rate at which deer can be hunted without causing their extinction? (a) The rate of growth remains proportional to the population while hunting decreases the population by a fixed amount (This assumes hunting is done continually throughout the year which is not what happens in real life) (b) The solution is u(t) = 7 e 7 t + 7 Solving u(t) = gives t = /7 log = 4694 years 7 /7 (c) We need the equilibrium u = b/7 to be less than the initial population so b < deer 8 (a) Prove that if u (t) and u (t) are any two distinct solutions to du = au with a > dt then u (t) u (t) as t (b) If a = and u () = u () = how long do you have to wait until u (t) u (t) >? (a) u (t) u (t) = e a t u () u () when a > (b) t = log(/)/ = 497 8 (a) Write down the exact solution to the initial value problem du dt = 7 u u() = evv 9/9/4 46 c 4 Peter J Olver
(b) Suppose you make the approximation u() = t what point does your solution differ from the true solution by unit? by? (c) nswer the same question if you also approximate the coefficient in the differential equation by du = 87 u dt (a) u(t) = e t/7 (b) One unit: t = log h /(/ ) i /(/7) = 68; units: t = log h /(/ ) i /(/7) = 68 (c) One unit: t 8 solves e t/7 e 87 t = solver) units: t 748 solves e t/7 e 87 t = (Use a numerical equation 8 Let a be complex Prove that u(t) = ce a t is the (complex) solution to our scalar ordinary differential equation (8) Describe the asymptotic behavior of the solution as t and the stability properties of the zero equilibrium solution The solution is still valid as a complex solution If Re a > then u(t) as t and the origin is an unstable equilibrium If Re a = then u(t) remains bounded t and the origin is a stable equilibrium If Re a < then u(t) as t and the origin is an asymptotically stable equilibrium ev 8 Eigenvalues and Eigenvectors 8 Find the eigenvalues and eigenvectors of the following matrices: (a) (b) (c) (d) (e) (i) (f ) 4 4 4 (j) (a) Eigenvalues: ; (b) Eigenvalues: (c) Eigenvalues: ; 6 4 (g) 6 6 (h) 7 4 4 4 (k) 4 eigenvectors: ; eigenvectors: 4 eigenvectors: (d) Eigenvalues: + i i ; eigenvectors: i (e) Eigenvalues: 4 ; eigenvectors: i evv 9/9/4 47 c 4 Peter J Olver
(f ) Eigenvalues: 6 + 6; eigenvectors: + i + i (g) Eigenvalues: + i i ; eigenvectors: i + i (h) Eigenvalues: ; eigenvectors: (i) simple eigenvalue eigenvector ; double eigenvalue eigenvectors (j) Two double eigenvalues: eigenvectors and 7 eigenvectors (k) Eigenvalues: 4; eigenvectors: cos θ sin θ 8 (a) Find the eigenvalues of the rotation matrix R θ = For what values sin θ cos θ of θ are the eigenvalues real? (b) Explain why your answer gives an immediate solution to Exercise 7c (a) The eigenvalues are cos θ ± i sin θ = e ± i θ with eigenvectors They are real i only for θ = and π (b) ecause R θ a I has an inverse if and only if a is not an eigenvalue 8 nswer Exercise 8a for the reflection matrix F θ = The eigenvalues are ± with eigenvectors ( sin θ cos θ ) T cos θ sin θ sin θ cos θ 84 Write down (a) a matrix that has as one of its eigenvalues and ( ) T as a corresponding eigenvector; (b) a matrix that has ( ) T as an eigenvector for the eigenvalue (a) O and (b) I are trivial examples 8 (a) Write out the characteristic equation for the matrix α β γ (b) Show that given any numbers a b and c there is a matrix with characteristic equation λ + aλ + bλ + c = (a) The characteristic equation is λ + aλ + bλ + c = (b) Use the matrix in part (a) 86 Find the eigenvalues and eigenvectors of the cross product matrix = c b c a b a The eigenvalues are ± i a + b + c If a = b = c = then = O and all vectors evv 9/9/4 48 c 4 Peter J Olver
are eigenvectors Otherwise the eigenvectors are i + i a b c b a i a + b + c 87 Find all eigenvalues and eigenvectors of the following complex matrices: (a) (b) (c) (d) i i i i + i + i (a) Eigenvalues: i + i ; eigenvectors: (b) Eigenvalues: ± ; eigenvectors: i ( ± ) (c) Eigenvalues: i ; eigenvectors: (d) simple eigenvector + i ; i double eigenvectors + i ac bc a + b + i i i i i + i + i + i 88 Find all eigenvalues and eigenvectors of (a) the n n zero matrix O; (b) the n n identity matrix I (a) Since Ov = = v we conclude that is the only eigenvalue; all nonzero vectors are eigenvectors (b) Since I v = v = v we conclude that is the only eigenvalue; all nonzero vectors are eigenvectors 89 Find the eigenvalues and eigenvectors of an n n matrix with every entry equal to Hint: Try with n = and then generalize For n = the eigenvalues are and the eigenvectors are and For n = the eigenvalues are and the eigenvectors are and In general the eigenvalues are with multiplicity n and n which is simple The eigenvectors corresponding to the eigenvalues are of the form ( v v v n ) T where v = and v j = for j = n The eigenvector corresponding to n is ( ) T 8 Let be a given square matrix (a) Explain in detail why any nonzero scalar multiple of an eigenvector of is also an eigenvector (b) Show that any nonzero linear combination of two eigenvectors v w corresponding to the same eigenvalue is also an eigenvector (c) Prove that a linear combination cv + dw with c d of two eigenvectors corresponding to different eigenvalues is never an eigenvector (a) If v = λv then (cv) = cv = cλv = λ(cv) and so cv satisfies the eigenvector equation with eigenvalue λ Moreover since v also cv for c and so cv is a bona fide eigenvector (b) If v = λv w = λw then (cv + dw) = cv + dw = cλv + dλw = λ(cv + dw) (c) Suppose v = λv w = µw Then v and w must be linearly independent as otherwise they would be scalar multiples of each other and hence have the same eigenvalue Thus (cv + dw) = cv + dw = cλv + dµw = ν(cv + dw) if and only if cλ = cν and dµ = dν which when λ µ is only possible if either c = or d = evv 9/9/4 49 c 4 Peter J Olver
8 True or false: If v is a real eigenvector of a real matrix then a nonzero complex multiple w = cv for c is a complex eigenvector of True by the same computation as in Exercise 8(a) cv is an eigenvector for the same (real) eigenvalue λ 8 Define the shift map S: n n by S v v v n v n T = ( v v v n v ) T (a) Prove that S is a linear map and write down its matrix representation (b) Prove that is an orthogonal matrix (c) Prove that the sampled exponential vectors ω ω n in (84) form an eigenvector basis of What are the eigenvalues? (a) = (b) T = I by direct computation or equivalently the columns of are the standard orthonormal basis vectors e n e e e n written in a slightly different order (c) Since ω k = S ω k = e k π i /n e 4 k π i /n e (n ) k π i /n«t e k π i /n e 4 k π i /n e (n ) k π i /n«t = e k π i /n ω k and so ω k is an eigenvector with corresponding eigenvalue e k π i /n 4 4 8 (a) ompute the eigenvalues and corresponding eigenvectors of = (b) ompute the trace of and check that it equals the sum of the eigenvalues (c) Find the determinant of and check that it is equal to to the product of the eigenvalues (a) Eigenvalues: ; eigenvectors: ( ) T T ( ) T (b) tr = = + + (c) det = = ( ) 84 Verify the trace and determinant formulae (84 ) for the matrices in Exercise 8 (a) tr = = + ( ); det = = ( ) (b) tr = 6 = + ; det = 6 = (c) tr = 4 = + ; det = 4 = (d) tr = = ( + i ) + ( i ); det = = ( + i ) ( i ) (e) tr = 8 = 4 + + ; det = = 4 (f ) tr = = + 6 + ( 6); det = 6 = 6 ( 6) (g) tr = = + ( + i ) + ( i ); det = = ( + i ) ( i ) (h) tr = 4 = + + ; det = = (i) tr = = ( ) + + ; det = 4 = ( ) (j) tr = = ( ) + ( ) + 7 + 7; det = 49 = ( ) ( ) 7 7 evv 9/9/4 4 c 4 Peter J Olver
(k) tr = = + + + 4; det = 4 = 4 8 (a) Find the explicit formula for the characteristic polynomial det( λ I ) = λ + aλ bλ + c of a general matrix Verify that a = tr c = det What is the formula for b? (b) Prove that if has eigenvalues λ λ λ then a = tr = λ + λ + λ b = λ λ + λ λ + λ λ c = det = λ λ λ (a) a = a + a + a = tr b = a a a a + a a a a + a a a a c = a a a + a a a + a a a a a a a a a a a a = det (b) When the factored form of the characteristic polynomial is multiplied out we obtain (λ λ )(λ λ )(λ λ ) = λ + (λ + λ + λ )λ (λ λ + λ λ + λ λ )λ λ λ λ giving the eigenvalue formulas for a b c 86 Prove that the eigenvalues of an upper triangular (or lower triangular) matrix are its diagonal entries If U is upper triangular so is U λ I and hence p(λ) = det(u λ I ) is the product of the diagonal entries so p(λ) = Q (u ii λ) and so the roots of the characteristic equation are the diagonal entries u u nn 87 Let J a be the n n Jordan block matrix (8) Prove that its only eigenvalue is λ = a and the only eigenvectors are the nonzero scalar multiples of the standard basis vector e Since J a λ I is an upper triangular matrix with λ a on the diagonal its determinant is det(j a λ I ) = (a λ) n and hence its only eigenvalue is λ = a of multiplicity n (Or use Exercise 86) Moreover (J a a I )v = ( v v v n ) T = if and only if v = ce 88 Suppose that λ is an eigenvalue of (a) Prove that cλ is an eigenvalue of the scalar multiple c (b) Prove that λ + d is an eigenvalue of + d I (c) More generally cλ + d is an eigenvalue of = c + d I for scalars c d parts (a) (b) are special cases of part (c): If v = λ v then v = (c + d I )v = (c λ + d) v 89 Show that if λ is an eigenvalue of then λ is an eigenvalue of If v = λv then v = λv = λ v and hence v is also an eigenvector of with eigenvalue λ 8 True or false: (a) If λ is an eigenvalue of both and then it is an eigenvalue of the sum + (b) If v is an eigenvector of both and then it is an eigenvector of + (a) False For example is an eigenvalue of both and but the eigenvalues of + = are ± i (b) True If v = λv and v = µv then (+)v = (λ + µ)v and so v is an eigenvector with eigenvalue λ + µ 8 True or false: If λ is an eigenvalue of and µ is an eigenvalue of then λµ is an eigenvalue of the matrix product = False in general but true if the eigenvectors coincide: If v = λv and v = µv then v = (λµ)v and so v is an eigenvector with eigenvalue λµ 8 Let and be n n matrices Prove that the matrix products and have the same eigenvalues If v = λv then w = λw where w = v Thus as long as w it evv 9/9/4 4 c 4 Peter J Olver
is an eigenvector of with eigenvalue λ However if w = then v = and so the eigenvalue is λ = which implies that is singular ut then so is which also has as an eigenvalue Thus every eigenvalue of is an eigenvalue of The converse follows by the same reasoning Note: This does not imply their null eigenspaces or kernels have the same dimension; compare Exercise 88 In anticipation of Section 86 even though and have the same eigenvalues they may have different Jordan canonical forms 8 (a) Prove that if λ is a nonzero eigenvalue of then /λ is an eigenvalue of (b) What happens if has as an eigenvalue? (a) Starting with v = λ v multiply both sides by and divide by λ to obtain v = (/λ) v Therefore v is an eigenvector of with eigenvalue /λ (b) If is an eigenvalue then is not invertible 84 (a) Prove that if det > then has at least one eigenvalue with λ > det < are all eigenvalues λ <? Prove or find a counter-example (b) If (a) If all λ j then so is their product λ λ n = det which is a contradiction (b) False = has eigenvalues while det = 8 Prove that is a singular matrix if and only if is an eigenvalue Recall that is singular if and only if ker {} ny v ker satisfies v = = v Thus ker is nonzero if and only if has a null eigenvector 86 Prove that every nonzero vector v R n is an eigenvector of if and only if is a scalar multiple of the identity matrix Let v w be any two linearly independent vectors Then v = λv and w = µw for some λ µ ut v + w is an eigenvector if and only if (v + w) = λv + µw = ν(v + w) which requires λ = µ = ν Thus v = λv for every v which implies = λ I 87 How many unit eigenvectors correspond to a given eigenvalue of a matrix? If λ is a simple real eigenvalue then there are two real unit eigenvectors: u and u For a complex eigenvalue if u is a unit complex eigenvector so is e i θ u and so there are infinitely many complex unit eigenvectors (The same holds for a real eigenvalue if we also allow complex eigenvectors) If λ is a multiple real eigenvalue with eigenspace of dimension greater than then there are infinitely many unit real eigenvectors in the eigenspace 88 True or false: (a) Performing an elementary row operation of type # does not change the eigenvalues of a matrix (b) Interchanging two rows of a matrix changes the sign of its eigenvalues (c) Multiplying one row of a matrix by a scalar multiplies one of its eigenvalues by the same scalar ll false Simple examples suffice to disprove them 89 (a) True or false: If λ v and λ v solve the eigenvalue equation (8) for a given matrix so does λ + λ v + v (b) Explain what this has to do with linearity False The eigenvalue equation v = λv is not linear in the eigenvalue and eigenvector since (v + v ) (λ + λ )(v + v ) in general 8 n elementary reflection matrix has the form Q = I uu T where u R n is a unit vector (a) Find the eigenvalues and eigenvectors for the elementary reflection matrices evv 9/9/4 4 c 4 Peter J Olver
corresponding to the following unit vectors: (i) (ii) 4 (iii) (iv) (b) What are the eigenvalues and eigenvectors of a general elementary reflection matrix? (a) (i) Q = (ii) Q = Eigenvalue has eigenvector: 7 4 7 4 4 Eigenvalue has eigenvector: ; Eigenvalue has eigenvector: 4 ; Eigenvalue has eigenvector: (iii) Q = Eigenvalue has eigenvector: ; Eigenvalue has eigenvectors: (iv) Q = Eigenvalue has eigenvector: ; Eigenvalue has eigenvectors: (b) u is an eigenvector with eigenvalue ll vectors orthogonal to u are eigenvectors with eigenvalue + 8 Let and be similar matrices so = S S for some nonsingular matrix S (a) Prove that and have the same characteristic polynomial: p (λ) = p (λ) (b) Explain why similar matrices have the same eigenvalues (c) Do they have the same eigenvectors? If not how are their eigenvectors related? (d) Prove that the converse is not true by showing that and have the same eigenvalues but are not similar det( λ I ) = det(s S λ I ) = det h S ( λ I )S i (a) = det S det( λ I ) det S = det( λ I ) (b) The eigenvalues are the roots of the common characteristic equation (c) Not usually If w is an eigenvector of then v = S w is an eigenvector of and conversely (d) oth have as a double eigenvalue Suppose = S S or equivalently S = S for some S = x y Then equating entries z w we must have x y = x x + y = z w = z + w = w which implies x = y = z = w = and so S = O which is not invertible 8 Let be a nonsingular n n matrix with characteristic polynomial p (λ) (a) Explain how to construct the characteristic polynomial p (λ) of its inverse directly from p (λ) 4 4 (b) heck your result when = (i) (ii) 4 (a) p (λ) = det( λ I ) = det λ λ I = ( λ)n det p λ evv 9/9/4 4 c 4 Peter J Olver
Or equivalently if p (λ) = ( ) n λ n + c n λ n + + c λ + c then since c = det " p (λ) = ( ) n λ n + c λ n + + c # n λ + c c c (b) (i) = Then p (λ) = λ λ while (ii) = p (λ) = λ + λ = λ 4 6 4 4 8 7 λ λ + Then p (λ) = λ + λ 7λ + while p (λ) = λ + 7 λ λ + = λ λ + λ 7 λ + 8 square matrix is called nilpotent if k = O for some k (a) Prove that the only eigenvalue of a nilpotent matrix is (The converse is also true; see Exercise 86) (b) Find examples where k O but k = O for k = and in general (a) If v = λ v then = k v = λ k v and hence λ k = (b) If has size (k + ) (k + ) and is all zero except for a ii+ = on the supra-diagonal ie a k k Jordan block J k with zeros along the diagonal 84 (a) Prove that every eigenvalue of a matrix is also an eigenvalue of its transpose T (b) Do they have the same eigenvectors? (c) Prove that if v is an eigenvector of with eigenvalue λ and w is an eigenvector of T with a different eigenvalue µ λ then v and w are orthogonal vectors with respect to the dot product (d) Illustrate this result when (i) = 4 (ii) = 4 (a) det( T λ I ) = det( λ I ) T = det( λ I ) and hence and T have the same characteristic polynomial (b) No See the examples (c) λ v w = (v) T w = v T T w = µ v w so if µ λ v w = and the vectors are orthogonal (d) (i) The eigenvalues are ; the eigenvectors of are v = v = ; the eigenvectors of T are w = w = and v w are orthogonal as are v w The eigenvalues are ; the eigenvectors of are v = v = v = ; the eigenvectors of T are w = w = w = Note that v i is orthogonal to w j whenever i j 8 (a) Prove that every real matrix has at least one real eigenvalue (b) Find a real evv 9/9/4 44 c 4 Peter J Olver
4 4 matrix with no real eigenvalues (c) an you find a real matrix with no real eigenvalues? (a) The characteristic equation of a matrix is a cubic polynomial and hence has at least one real root (b) has eigenvalues ± i (c) No since the characteristic polynomial is degree and hence has at least one real root 86 (a) Show that if is a matrix such that 4 = I then the only possible eigenvalues of are i and i (b) Give an example of a real matrix that has all four numbers as eigenvalues (a) If v = λv then v = 4 v = λ 4 v and hence any eigenvalue must satisfy λ 4 = (b) 87 projection matrix satisfies P = P Find all eigenvalues and eigenvectors of P If P v = λv then P v = λ v Since P v = P v we find λv = λ v and so since v = λ so the only eigenvalues are λ = ll v rng P are eigenvectors with eigenvalue since if v = P u then P v = P u = P u = v whereas all w ker P are null eigenvectors 88 True or false: ll n n permutation matrices have real eigenvalues False For example has eigenvalues ± i 89 (a) Show that if all the row sums of are equal to then has as an eigenvalue (b) Suppose all the column sums of are equal to Does the same result hold? Hint: Use Exercise 84 (a) ccording to Exercise 9 if z = ( ) T then z is the vector of row sums of and hence by the assumption z = z so z is an eigenvector with eigenvalue (b) Yes since the column sums of are the row sums of T and Exercise 84 says that and T have the same eigenvalues 84 Let Q be an orthogonal matrix (a) Prove that if λ is an eigenvalue then so is /λ (b) Prove that all its eigenvalues are complex numbers of modulus λ = In particular the only possible real eigenvalues of an orthogonal matrix are ± (c) Suppose v = x + i y is a complex eigenvector corresponding to a non-real eigenvalue Prove that its real and imaginary parts are orthogonal vectors having the same Euclidean norm (a) If Qv = λv then Q T v = Q v = λ v and so λ is an eigenvalue of Q T Exercise 84 says that a matrix and its transpose have the same eigenvalues (b) If Qv = λv then by Exercise 6 v = Qv = λ v and hence λ = Note that this proof also applies to complex eigenvalues/eigenvectors using the Hermitian norm (c) If e i θ = cos θ + i sin θ is the eigenvalue then Qx = cos θx sin θy Qy = sin θx + cos θy evv 9/9/4 4 c 4 Peter J Olver
while Q x = cos θx sin θy Q y = sin θx + cos θy Thus cos θ x + cos θ sin θx y+sin θy = Q x = x = Qx = cos θ x cos θ sin θx y+sin θy and so for x y = provided θ ± π ±π Moreover x = cos θ x + sin θy while by the same calculation y = sin θ x + cos θy which imply x = y For θ = π we have Qx = y Qy = x and so x y = (Qy) (Qx) = x y is also zero 84 (a) Prove that every proper orthogonal matrix has + as an eigenvalue or false: n improper orthogonal matrix has as an eigenvalue (b) True (a) ccording to Exercise 8 a orthogonal matrix has at least one real eigenvalue which by Exercise 84 must be ± If the other two eigenvalues are complex conjugate µ± i ν then the product of the eigenvalues is ±(µ +ν ) Since this must equal the determinant of Q which by assumption is positive we conclude that the real eigenvalue must be + Otherwise all the eigenvalues of Q are real and they cannot all equal as otherwise its determinant would be negative (b) True It must either have three real eigenvalues of ± of which at least one must be as otherwise its determinant would be + or a complex conjugate pair of eigenvalues λ λ and its determinant is = ± λ so its real eigenvalue must be and its complex eigenvalues ± i 84 (a) Show that the linear transformation defined by proper orthogonal matrix corresponds to rotating through an angle around a line through the origin in R the axis of the rotation Hint: Use Exercise 84(a) (b) Find the axis and angle of rotation of the orthogonal matrix 4 48 6 4 6 6 (a) The axis of the rotation is the eigenvector v corresponding to the eigenvalue + Since Qv = v the rotation fixes the axis and hence must rotate around it hoosing an orthonormal basis u u u where u is a unit eigenvector in the direction of the axis of rotation while u + i u is a complex eigenvector for the eigenvalue e i θ In this basis Q has matrix form cos θ sin θ where θ is the angle of rotation (b) The axis is the eigenvec- sin θ cos θ tor for the eigenvalue The complex eigenvalue is 7 + i and so the angle is θ = cos 7 9 84 Find all invariant subspaces cf Exercise 74 of a rotation in R In general besides the trivial invariant subspaces {} and R the axis of rotation and its orthogonal complement plane are invariant If the rotation is by 8 then any line in the orthogonal complement plane is also invariant If R = I then every subspace is invariant 844 Suppose Q is an orthogonal matrix (a) Prove that K = I Q Q T is a positive semi-definite matrix (b) Under what conditions is K >? (a) (Q I ) T (Q I ) = Q T Q Q Q T + I = I Q Q T = K and hence K is a Gram matrix which is positive semi-definite by Theorem 8 (b) The Gram matrix is positive definite if and only if ker(q I ) = {} which means that evv 9/9/4 46 c 4 Peter J Olver
Q does not have an eigenvalue of 84 Prove that every proper affine isometry F (x) = Q x + b of R where det Q = + is one of the following: (a) a translation x + b (b) a rotation centered at some point of R or (c) a screw consisting of a rotation around an axis followed by a translation in the direction of the axis Hint: Use Exercise 84 If Q = I then we have a translation Otherwise we can write F (x) = Q(x c) + c in the form of a rotation around the center point c provided we can solve (Q I )c = b y the Fredholm alternative this requires b to be orthogonal to coker(q I ) which is also spanned by the rotation axis v ie the eigenvector for the eigenvalue + of Q T = Q More generally we write F (x) = Q(x c) + c + tv and identify the affine map as a screw around the axis in the direction of v passing through c 846 Let M n be the n n tridiagonal matrix hose diagonal entries are all equal to and whose sub- and super-diagonal entries all equal (a) Find the eigenvalues and eigenvectors of M and M directly (b) Prove that the eigenvalues and eigenvectors of M n are explicitly given by λ k = cos k π n + v k = sin k π n + How do you know that there are no other eigenvalues? (a) For M = : eigenvalues ; eigenvectors k π nk π T sin sin k = n n + n + eigenvalues ; eigenvectors (b) The j th entry of the eigenvalue equation M n v k = λ k v k reads sin (j )k π n + + sin (j + )k π n + = cos For M = k π n + sin j k π n + : which is a standard trigonometric identity: sin α + sin β = cos α β sin α + β These are all the eigenvalues because an n n matrix has at most n eigenvalues 847 Let a b be fixed scalars Determine the eigenvalues and eigenvectors of the n n tridiagonal matrix with all diagonal entries equal to a and all sub- and super-diagonal entries equal to b Hint: See Exercises 88 46 We have = a I +bm n so by Exercises 88 and 846 it has the same eigenvectors k π as M n while its corresponding eigenvalues are a + bλ k = a + b cos for k = n n + 848 Find a formula for the eigenvalues of the tricirculant n n matrix Z n that has s on the sub- and super-diagonals as well as its ( n) and (n ) entries while all other entries are Hint: Use Exercise 846 as a guide λ k = cos k π n v k π 4k π 6k π (n )k π T k = cos cos cos cos n n n n for k = n 849 Let be an n n matrix with eigenvalues λ λ k and an m m matrix with evv 9/9/4 47 c 4 Peter J Olver
eigenvalues µ µ l Show that the (m+n) (m+n) block diagonal matrix D = O O has eigenvalues λ λ k µ µ l and no others How are the eigenvectors related? Note first that if v = λv then D v = v = λ v v and so is an eigenvector for D with eigenvalue λ Similarly each eigenvalue µ and eigenvector w of gives an eigenvector of D Finally to check that D has no other eigenvalue we compute w D v = v v = λ and hence if v then λ is an eigenvalue of while if w w w w then it must also be an eigenvalue for 8 Let = a b be a matrix (a) Prove that satisfies its own characteristic c d equation meaning p () = (tr ) + (det ) I = O Remark: This result is a special case of the ayley Hamilton Theorem to be developed in Exercise 866 (b) Prove the inverse formula (tr ) I = when det (c) heck the ayley det Hamilton and inverse formulas when = (a) Follows by direct computation (b) Multiply the characteristic equation by and rearrange terms (c) tr = 4 det = 7 and one checks 4 + 7 I = O 8 Deflation: Suppose has eigenvalue λ and corresponding eigenvector v (a) Let b be any vector Prove that the matrix = v b T also has v as an eigenvector now with eigenvalue λ β where β = v b (b) Prove that if µ λ β is any other eigenvalue of then it is also an eigenvalue of Hint: Look for an eigenvector of the form w + cv where w is the eigenvector of (c) Given a nonsingular matrix with eigenvalues λ λ λ n and λ λ j j explain how to construct a deflated matrix whose eigenvalues are λ λ n (d) Try out your method on the matrices and (a) v = ( v b T )v = v (b v)v = (λ β)v (b) (w + cv) = ( v b T )(w + cv) = µw + (c(λ β) b w)v = µ(w + cv) provided c = b w/(λ β µ) (c) Set = λ v b T where v is the first eigenvector of and b is any vector such that b v = For example we can set b = v / v (Weilandt deflation [] chooses b = r j /(λ v j ) where v j is any nonzero entry of v and r j is the corresponding row of ) (d) (i) The eigenvalues of are 6 and the eigenvectors The deflated matrix = λ v vt v = has eigenvalues and eigenvectors (ii) The eigenvalues of are 4 and the eigenvectors The de- evv 9/9/4 48 c 4 Peter J Olver
flated matrix = λ v vt v = eigenvectors 4 4 has eigenvalues and the diag 8 Eigenvector ases and Diagonalization 8 Which of the following are complete eigenvalues for the indicated matrix? What is the dimension of the associated eigenspace? (a) (b) 4 4 (c) (d) (e) 4 (f ) i i i (g) i (a) omplete dim = with basis ( ) T (b) Not complete dim = with basis ( ) T (c) omplete dim = with basis ( ) T (d) Not an eigenvalue (e) omplete dim = with basis ( ) T ( ) T (f ) omplete dim = with basis ( i ) T (g) Not an eigenvalue (h) Not complete dim = with basis ( ) T (h) 8 Find the eigenvalues and a basis for the each of the eigenspaces of the following matrices Which are complete? 4 4 4 6 8 i (a) (b) (c) (d) (e) 4 6 4 i 6 8 (f ) 4 4 (g) 6 (h) (i) 4 4 6 4 (a) Eigenvalue: Eigenvector: (b) Eigenvalues: Eigenvectors: Not complete omplete evv 9/9/4 49 c 4 Peter J Olver
± i (c) Eigenvalues: ± i Eigenvectors: omplete i i (d) Eigenvalues: i Eigenvectors: omplete (e) Eigenvalue has eigenspace basis Not complete (f ) Eigenvalue has eigenspace basis Eigenvalue has omplete (g) Eigenvalue has eigenspace basis Eigenvalue has Not complete (h) Eigenvalue has Eigenvalue has Eigenvalue has omplete (i) Eigenvalue has eigenspace basis Eigenvalue has Not complete 8 Which of the following matrices admit eigenvector bases of R n? For those that do exhibit such a basis If not what is the dimension of the subspace of R n spannedby the eigenvectors? (a) (b) (c) (d) 4 4 (e) (f ) (g) (h) 4 (a) Eigenvalues: 4; the eigenvectors (b) Eigenvalues: i + i ; the eigenvectors is i form a basis for R i are not real so the dimension (c) Eigenvalue: ; there is only one eigenvector v = spanning a one-dimensional subspace of R (d) The eigenvalue has eigenvector while the eigenvalue has eigenvectors The eigenvectors form a basis for R (e) The eigenvalue has eigenvector while the eigenvalue has eigenvector The eigenvectors span a two-dimensional subspace of R evv 9/9/4 4 c 4 Peter J Olver
8 (f ) λ = The eigenvectors are and forming a basis for R 7 i i (g) the eigenvalues are i i The eigenvectors are and The real eigenvectors span only a one-dimensional subspace of R 4 (h) The eigenvalues are i i + The eigenvectors are i i 6 The real eigenvectors span a two-dimensional subspace of R 4 84 nswer Exercise 8 with R n replaced by n ases (abdfgh) all have eigenvector bases of n 8 (a) Give an example of a matrix that only has as an eigenvalue and has only one linearly independent eigenvector (b) Give an example that has two linearly independent eigenvectors (a) (b) 86 True or false: (a) Every diagonal matrix is complete (b) Every upper triangular matrix is complete (a) True The standard basis vectors are eigenvectors (b) False The Jordan matrix is incomplete since e is the only eigenvector 87 Prove that if is a complete matrix so is c+d I where c d are any scalars Hint: Use Exercise 88 ccording to Exercise 88 every eigenvector of is an eigenvector of c + d I with eigenvalue cλ + d and hence if has a basis of eigenvectors so does c + d I 88 (a) Prove that if is complete so is (b) Give an example of an incomplete matrix such that is complete (a) Every eigenvector of is an eigenvector of with eigenvalue λ and hence if has a basis of eigenvectors so does (b) = with = O 89 Suppose v v n forms an eigenvector basis for the complete matrix with λ λ n the corresponding eigenvalues Prove that every eigenvalue of is one of the λ λ n Suppose v = λv Write v = nx i = c i v i Then v = independence λ i c i = λ c i THus either λ = λ i or c i = nx i = i i c i λ i v i and hence by linear 8 (a) Prove that if λ is an eigenvalue of then λ n is an eigenvalue of n (b) State and prove a converse if is complete Hint: Use Exercise 89 (The completeness hypothesis is not essential but this is harder relying on the Jordan canonical form) QED (a) If v = λv then by induction n v = λ n v and hence v is an eigenvector with eigenvalue λ n (b) onversely if is complete and n has eigenvalue µ then there is a (complex) n th root λ = n µ that is an eigenvalue of Indeed the eigenvector basis of is an evv 9/9/4 4 c 4 Peter J Olver
eigenvector basis of n and hence using Exercise 89 every eigenvalue of n is the n th power of an eigenvalue of 8 Show that if is complete then any similar matrix = S S is also complete s in Exercise 8 if v is an eigenvector of then S v is an eigenvector of Moreover if v v n form a basis so do S v S v n ; see Exercise 4 for details 8 Let U be an upper triangular matrix with all its diagonal entries equal Prove that U is complete if and only if U is a diagonal matrix ccording to Exercise 86 its only eigenvalue is λ the common value of its diagonal entries and so all eigenvectors belong to ker(u λ I ) Thus U is complete if and only if dim ker(u λ I ) = n if and only if U λ I = O QED 8 Show that each eigenspace of an n n matrix is an invariant subspace as defined in Exercise 74 Let V = ker( λ I ) If v V then v V since ( λ I )v = ( λ I )v = QED 84 (a) Prove that if v = x ± i y is a complex conjugate pair of eigenvectors of a real matrix corresponding to complex conjugate eigenvalues µ ± i ν with ν then x and y are linearly independent real vectors (b) More generally if v j = x j ± i y j j = k are complex conjugate pairs of eigenvectors corresponding to distinct pairs of complex conjugate eigenvalues µ j ± i ν j ν j then the real vectors x x k y y k are linearly independent (a) Let µ ± i ν be the corresponding eigenvalues Then the complex eigenvalue equation (x + i y) = (µ + i ν)(x + i y) implies that x = µx ν y and y = ν x + µy Now suppose cx + dy = for some (c d) ( ) Then = (cx + dy) = (cµ + dν)x + ( cν + dµ)y The determinant of the coefficient matrix of these two sets of equations for x y is c d det = (c cµ + dν cν + dµ + d )ν because we are assuming the eigenvalues are truly complex This implies x = y = which contradicts the fact that we have an eigenvector (b) This is proved by induction Suppose we know x x k y y k are linearly independent If x x k y y k were linearly dependent there would exist (c k d k ) ( ) such that z k = c k x k + d k y k is some linear combination of x x k y y k ut then z k = (c k µ + d k ν)x k + ( c k ν + d k µ)y k is also a linear combination as is z k = (c k (µ ν ) + d k µν)x k + ( c k µν + d k (µ ν ))y k Since the coefficient matrix of the first two such vectors is nonsingular a suitable linear combination would vanish: = az k + bz k + c z k which would give a vanishing linear combination of x x k y y k which by the induction hypothesis must be trivial little more work demonstrates that this implies z k = and so in contradiction to part (b) would imply x k y k are linearly dependent QED 8 Diagonalize the following matrices: (a) 9 (b) 6 4 (c) 4 evv 9/9/4 4 c 4 Peter J Olver
(d) (h) (a) S = (e) (i) 8 (f ) (j) D = (b) S = D = (c) S = + i i D = + i i (d) S = D = (e) S = 6 D = 7 7 i + i (f ) S = D = (g) S = (h) S = (i) S = (j) S = 6 (g) 8 7 + i i D = 4 6 D = D = i i i + i + i i D = i i 86 Diagonalize the Fibonacci matrix F = = + 87 Diagonalize the matrix the result? S = i i + + of rotation through 9 How would you interpret D = i rotation does not stretch any real vectors but i evv 9/9/4 4 c 4 Peter J Olver
somehow corresponds to two complex stretches 88 Diagonalize the rotation matrices (a) (a) (b) = = i i i i (b) i i i i + i i i i 89 Which of the following matrices have real diagonal forms? (a) (b) 4 8 (c) (d) (e) (f ) 4 (a) Yes distinct real eigenvalues (b) No complex eigenvalues ± i 6 (c) No complex eigenvalues ± i (d) No incomplete eigenvalue (and complete eigenvalue ) (e) Yes distinct real eigenvalues 4 (f ) Yes complete real eigenvalues 8 Diagonalize the following complex matrices: (a) (b) (c) i i i i + i (a) S = D = + i + i (b) S = + i D = i + i (c) S = i D = 4 (d) S = i + i D = i i i + i (d) i + i i i i 8 Write down a real matrix that has (a) eigenvalues and corresponding eigenvectors (b) eigenvalues and associated eigenvectors ; (c) an eigenvalue of and corresponding eigenvectors ; (d) an eigenvalue evv 9/9/4 44 c 4 Peter J Olver
+ i and corresponding eigenvector ; (e) an eigenvalue and corresponding eigenvector + i i ; (f ) an eigenvalue + i and corresponding eigenvector i i Use = S ΛS For parts (e) (f ) you can choose any other eigenvalues and eigenvectors you want to fill in S and Λ (a) (b) (c) 6 6 7 4 8 6 6 4 (d) 4 4 (e) example: (f ) example: 6 8 matrix has eigenvalues and and associated eigenvectors and Write down the matrix form of the linear transformation L[u] = u in terms of (a) the standard basis e e ; (b) the basis consisting of its eigenvectors; (c) the basis 4 (a) 6 (b) 8 (c) 4 6 8 Prove that two complete matrices have the same eigenvalues (with multiplicities) if and only if they are similar ie = S S for some nonsingular matrix S Let S be the eigenvector matrix for and S the eigenvector matrix for Thus by the hypothesis S S = Λ = S S and hence = S S S S = S S where S = S S 84 Let be obtained from by permuting both its rows and columns using the same permutation π so b ij = a π(i)π(j) Prove that and have the same eigenvalues How are their eigenvectors related? Let v = ( v v v n ) T be an eigenvector for Let ev be obtained by applying the permutation to the entries of v so ev i = v π(i) Then the i th entry of ev is nx j = b ij ev j = nx j = a π(i)π(j) v π(j) = nx j = and hence ev is an eigenvector of with eigenvalue λ a π(i)j v j = λv π(i) = λev i QED 8 True or false: If is a complete upper triangular matrix then it has an upper triangular eigenvector matrix S True Let λ j = a jj denote the j th diagonal entry of which is the same as the j th eigenvalue We will prove that the corresponding eigenvector is a linear combination of e e j which is equivalent to the eigenvector matrix S being upper triangular We use induction on the size n Since is upper triangular it leaves the subspace V spanned by e e n invariant and hence its restriction is an (n ) (n ) upper triangular matrix Thus by induction and completeness possesses n eigenvectors of the required form The remaining eigenvector v n cannot belong to V (otherwise the eigenvectors would be linearly dependent) and hence must involve e n QED 86 Suppose the n n matrix is diagonalizable How many different diagonal forms does it have? evv 9/9/4 4 c 4 Peter J Olver
The diagonal entries are all eigenvalues and so are obtained from each other by permutation If all eigenvalues are distinct then there are n different diagonal forms otherwise n if it has distinct eigenvalues of multiplicities j j k there are distinct diagonal j j k forms 87 haracterize all complete matrices that are their own inverses: = Write down a non-diagonal example = I if and only if D = I and so all its eigenvalues are ± Examples: = with eigenvalues and eigenvectors ; or even simpler = 4 88 Two n n matrices are said to be simultaneously diagonalizable if there is a nonsingular matrix S such that both S S and S S are diagonal matrices (a) Show that simultaneously diagonalizable matrices commute: = (b) Prove that the converse is valid provided one of the matrices has no multiple eigenvalues (c) Is every pair of commuting matrices simultaneously diagonalizable? (a) If = S ΛS and = S D S where Λ D are diagonal then = S ΛD S = S D ΛS = since diagonal matrices commute (b) ccording to Exercise (e) the only matrices that commute with an n n diagonal matrix with distinct entries is another diagonal matrix Thus if = and = S ΛS where all entries of Λ are distinct then D = S S commutes with Λ and hence is a diagonal matrix (c) No the matrix commutes with the identity matrix but is not diagonalizable See also Exercise 4 evsym 84 Eigenvalues of Symmetric Matrices 84 Find the eigenvalues and an orthonormal eigenvector basis for the following symmetric matrices: 4 6 4 6 (a) (b) (c) (d) (e) 4 6 6 7 4 (a) eigenvalues: ; (b) eigenvalues: 7 ; eigenvectors: eigenvectors: 7 + (c) eigenvalues: 7 ; eigenvectors: q 6 6 q 6 + 6 + evv 9/9/4 46 c 4 Peter J Olver
(d) eigenvalues: 6 4; (e) eigenvalues: 9 ; eigenvectors: eigenvectors: 4 6 6 6 4 4 84 Determine whether the following symmetric matrices are positive definite by computing their eigenvalues Validate your conclusions by using the methods from hapter 4 4 (a) (b) (c) (d) 4 6 4 (a) eigenvalues ± 7; positive definite (b) eigenvalues 7; not positive definite (c) eigenvalues ; positive semi-definite (d) eigenvalues 6 ± ; positive definite 84 Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negative Use the fact that K = N is positive definite and so has all positive eigenvalues The eigenvalues of N = K are λ j where λ j are the eigenvalues of K lternatively mimic the proof in the book for the positive definite case 844 How many orthonormal eigenvector bases does a symmetric n n matrix have? If all eigenvalues are distinct there are n different bases governed by the choice of sign in the unit eigenvectors ±u k If the eigenvalues are repeated there are infinitely many since any orthonormal basis of each eigenspace will contribute to an orthonormal eigenvector basis of the matrix 84 Let = a b (a) Write down necessary and sufficient conditions on the entries c d a b c d that ensures that has only real eigenvalues (b) Verify that all symmetric matrices satisfy your conditions (a) The characteristic equation p(λ) = λ (a + d)λ + (ad bc) = has real roots if and only if its discriminant is non-negative: (a + d) 4(ad bc) = (a d) + 4bc which is the necessary and sufficient condition for real eigenvalues (b) If is symmetric then b = c and so the discriminant is (a d) + 4b 846 Let T = be a real skew-symmetric n n matrix (a) Prove that the only possible real eigenvalue of is λ = (b) More generally prove that all eigenvalues λ of are purely imaginary ie Re λ = (c) Explain why is an eigenvalue of whenever n is odd (d) Explain why if n = the eigenvalues of O are i ω i ω for some real ω (e) Verify these facts for the particular matrices (i) (ii) 4 (iii) (iv) 4 evv 9/9/4 47 c 4 Peter J Olver
(a) If v = λv and v is real then λ v = (v) v = (v) T v = v T T v = v T v = v (v) = λ v and hence λ = (b) Using the Hermitian dot product λ v = (v) v = v T T v = v T v = v (v) = λ v and hence λ = λ so λ is purely imaginary (c) Since det = cf Exercise 9 at least one of the eigenvalues of must be c b (d) The characteristic polynomial of = c a is λ + λ(a + b + c ) and b a hence the eigenvalues are ± i a + b + c and so are all zero if and only if = O (e) The eigenvalues are: (i) ± i (ii) ± i (iii) ± i (iv) ± i ± i 847 (a) Prove that every eigenvalue of a Hermitian matrix so T = as in Exercise 649 is real (b) Show that the eigenvectors corresponding to distinct eigenvalues are orthogonal under the Hermitian dot product on n (c) Find the eigenvalues and eigenvectors of the following Hermitian matrices and verify orthogonality: i i i (i) (ii) (iii) i i i + i i (a) Let v = λv Using the Hermitian dot product λ v = (v) v = v T T v = v T v = v (v) = λ v and hence λ = λ which implies that the eigenvalue λ is real (b) Let v = λv w = µw Then λv w = (v) w = v T T w = v T w = v (w) = µv w since µ is real Thus if λ µ then v w = (c) (i) eigenvalues ± ; eigenvectors: ( ) i ( + ) i i + i (ii) eigenvalues 4 ; eigenvectors: (iii) eigenvalues ± ; eigenvectors: i i 848 Let M > be a fixed positive definite n n matrix nonzero vector v is called a generalized eigenvector of the n n matrix K if K v = λm v v (8) where the scalar λ is the corresponding generalized eigenvalue (a) Prove that λ is a generalized eigenvalue of the matrix K if and only if it is an ordinary eigenvalue of the matrix M K How are the eigenvectors related? (b) Now suppose K is a symmetric matrix Prove that its generalized eigenvalues are all real Hint: First explain why this does not follow from part (a) Instead mimic the proof of part (a) of Theorem 8 using the weighted Hermitian inner product v w = v T M w in place of the dot product (c) Show that if K > then its generalized eigenvalues are all positive: λ > (d) Prove that the eigenvectors corresponding to different generalized eigenvalues are orthogonal under the weighted inner product v w = v T M w (e) Show that if the matrix pair K M has n distinct generalized eigenvalues then the eigenvectors form an orthogonal basis for R n Reevv 9/9/4 48 c 4 Peter J Olver
mark: One can by mimicking the proof of part (c) of Theorem 8 show that this holds even when there are repeated generalized eigenvalues (a) Rewrite (8) as M K v = λv and so v is an eigenvector for M K with eigenvalue λ (b) If v is an generalized eigenvector then since K M are real matrices K v = λ M v Therefore λ v = λv T M v = (λm v) T v = (K v) T v = v T (K v) = λv T M v = λ v and hence λ is real (c) If K v = λm v K w = µm w with λ µ and v w real then λ v w = (λm v) T w = (K v) T w = v T (K w) = µv T M w = µ v w and so if λ µ then v w = proving orthogonality (d) If K > then λ v v = v T (λm v) = v T K v > and so by positive definiteness of M λ > (e) Part (b) proves that the eigenvectors are orthogonal with respect to the inner product induced by M and so the result follows immediately from Theorem 849 ompute the generalized eigenvalues and eigenvectors as in (8) for the following matrix pairs Verify orthogonality of the eigenvectors under the appropriate inner product (a) K = M = (b) K = M = (c) K = (e) K = M = 4 8 M = (d) K = (f ) K = 6 8 8 4 6 M = 6 99 M = 9 4 9 (a) eigenvalues: ; eigenvectors: ; (b) eigenvalues: ; eigenvectors: ; (c) eigenvalues: 7 ; eigenvectors: ; 6 6 (d) eigenvalues: 9 ; eigenvectors: ; 4 (e) eigenvalues: ; eigenvectors: ; (f ) is a double eigenvalue with eigenvector basis while is a simple eigenvalue with eigenvector For orthogonality you need to select an M orthogonal basis of the two-dimensional eigenspace say by using Gram Schmidt 84 Let L = L : R n R n be a self-adjoint linear transformation with respect to the inner product Prove that all its eigenvalues are real and the eigenvectors are orthogonal Hint: Mimic the proof of Theorem 8 replacing the dot product by the given inner product evv 9/9/4 49 c 4 Peter J Olver
If L[v ] = λv then using the inner product λ v = L[v ] v = v L[v ] = λ v which proves that the eigenvalue λ is real Similarly if L[w ] = µ : w then λ v w = L[v ] w = v L[w ] = µ v w and so if λ µ then v w = 84 The difference map : n n is defined as = S I where S is the shift map of Exercise 8 (a) Write down the matrix corresponding to (b) Prove that the sampled exponential vectors ω ω n from (84) form an eigenvector basis of What are the eigenvalues? (c) Prove that K = T has the same eigenvectors as What are its eigenvalues? (d) Is K positive definite? (e) ccording to Theorem 8 the eigenvectors of a symmetric matrix are real and orthogonal Use this to explain the orthogonality of the sampled exponential vectors ut why aren t they real? (a) (b) Using Exercise 8(c) ω k = (S I )ω k = e k π i /n ω k and so ω k is an eigenvector of with corresponding eigenvalue e k π i /n (c) Since S is an orthogonal matrix S T = S and so S T ω k = e k π i /n ω k Therefore K ω k = (S T I )(S I )ω k = ( I S S T )ω k = e k π i /n e k π i /n ω k = cos k π i n ω k and hence ω k is an eigenvector of K with corresponding eigenvalue cos k π i n (d) Yes K > since its eigenvalues are all positive; or note that K = T is a Gram matrix with ker = {} (e) Each eigenvalue cos k π i (n k)π i = cos for k n n n is double with a twodimensional eigenspace spanned by ω k and ω n k = ω k The correpsonding real eigenvectors are Re ω k = ω k + ω n k and Im ω k = i ω k i ω n k On the other hand if k = n (which requires that n be even) the eigenvector ω n/ = ( )T is real c c c c c n c n c c c c n 84 n n n circulant matrix has the form = c n c n c c c n c c c c 4 c in which the entries of each succeeding row are obtained by moving all the previous row s entries one slot to the right the last entry moving to the front (a) heck that the shift matrix S of Exercise 8 the difference matrix and its symmetric product K of Exercise 84 are all circulant matrices (b) Prove that the sampled exponential vectors evv 9/9/4 44 c 4 Peter J Olver
ω ω n cf (84) are eigenvectors of Thus all circulant matrices have the same eigenvectors What are the eigenvalues? (c) Prove that Ω n Ω n = Λ where Ω n is the Fourier matrix in Exercise 76 and Λ is the diagonal matrix with the eigenvalues of along the diagonal (d) Find the eigenvalues and eigenvectors of the following circulant matrices: (i) (ii) (iii) (iv) (e) Find the eigenvalues of the tricirculant matrices in Exercise 7 an you find a general formula for the n n version? Explain why the eigenvalues must be real and positive Does your formula reflect this fact? (f ) Which of the preceding matrices are invertible? Write down a general criterion for checking the invertibility of circulant matrices (a) The eigenvector equation ω k = (c + c e k π i /n + c e 4 k π i /n + + +c n e (n ) k π i /n )ω k can either be proved directly or by noting that and using Exercise 8(c) (b) (i) Eigenvalues ; = c I + c S + c S + + c n S n eigenvectors (ii) Eigenvalues 6 i + i ; eigenvectors (iii) Eigenvalues i + i ; eigenvectors i i + i i i i i i + i i (iv) Eigenvalues 4 ; eigenvectors i i (c) The eigenvalues are (i) 6 ; (ii) 6 4 4 ; (iii) 6 7+ 7+ 7 7 ; (iv) in the n n case they are 4 + cos k π for k = n The eigenvalues are real and n positive because the matrices are symmetric positive definite (d) ases (iii) in (d) and all matrices in part (e) are invertible In general an n n circulant matrix is invertible if and only if none of the roots of the polynomial c +c x+ + c n x n = is an n th root of unity: x e k π/n 84 Write out the spectral decomposition of thefollowing matrices: 4 (a) (b) (c) (d) 4 4 evv 9/9/4 44 c 4 Peter J Olver