(I + T )’1 ’ I + T ¤ T (1 ’ T )’1 .

2

Conclude that

˜(I + T ) = I ’ T + (T ) T ,

where : L(U, U ) ’ L(U, U ) is such that (T ) ’ 0 as T ’ 0. Con-

clude that ˜ is di¬erentiable at I. [If you wish to con¬ne yourself to ¬nite

dimensional spaces, take U ¬nite dimensional, but there is no need.]

(vi) Show that ˜ is everywhere di¬erentiable on E with D˜(B) = ¦B ,

where ¦B : L(U, U ) ’ L(U, U ) is the linear map given by

¦B (S) = ’B ’1 SB ’1 .

Check that this reduces to a known result when dim U = 1.

588 A COMPANION TO ANALYSIS

Exercise K.282. (Spectral radius.) [13.1, T, ‘ ] We continue with the

ideas and notation of Exercise K.281.

(i) Suppose that A ∈ L(U, U ). Show that

Ajk+r ¤ Aj k

A r.

= inf n≥1 An 1/n

(ii) Continuing with the hypotheses of (i), show that

exists and, by using the result of (i), or otherwise that

An 1/n

’ .

We call the spectral radius of A and write ρ(A) = .

(iii) Show that ∞ An converges in the operator norm if ρ(A) < 1 and

n=0

diverges if ρ(A) > 1.

(iv) If ρ(A) < 1 show that I ’ A is invertible and (I ’ A)’1 = ∞ An .

n=0

Exercise K.283. [13.1, T] We continue with the ideas of Exercise K.282.

(i) If ± ∈ L(U, U ) and » is a scalar show that ρ(»±) = |»|ρ(±).

(ii) Write down a linear map β : Cm ’ Cm such that β m’1 = 0 but

β m = 0. What is the value of ρ(β)?

(iii) If the linear map ± : Cm ’ Cm has m distinct eigenvalues explain

brie¬‚y why we can ¬nd an invertible linear map θ : Cm ’ Cm such that θ±θ ’1

has a diagonal matrix D with respect to the standard basis. By considering

the behaviour of D n , or otherwise, show that

ρ(±) = max{|»| : » an eigenvalue of ±}.

(For an improvement of this result see Exercise K.285.)

(iv) Give two linear maps ±, β : C2 ’ C2 such that ρ(±) = ρ(β) = 0 but

ρ(± + β) = 1.

Exercise K.284. (Cayley-Hamilton.) This question requires you to know

that any linear map ± : Cm ’ Cm has an upper triangular matrix with re-

spect to some basis. Recall also that, if ± : Cm ’ Cm we can de¬ne its

characteristic polynomial χ± by the condition det(tι ’ ±) = χ± (t) for all

t ∈ C. (Here and elsewhere, ι : Cm ’ Cm is the identity map.)

(i) We work with the standard basis on Cm . Explain why, if ± : Cm ’ Cm

is linear, we can ¬nd an invertible linear map θ : Cm ’ Cm such that

β = θ±θ’1 has an upper triangular matrix with respect to the standard

basis.

(ii) Show that, if β : Cm ’ Cm has an upper triangular matrix with

respect to the standard basis and > 0, then we can ¬nd γ : Cm ’ Cm with

m distinct eigenvalues such that γ ’ β < .

589

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) Deduce that, if ± : Cm ’ Cm is linear we can ¬nd linear ±k : Cm ’

Cm with m distinct eigenvalues and

±k ’ ± ’ 0 as k ’ ∞.

(iv) (Pure algebra) If ± : Cm ’ Cm has n distinct eigenvalues, show, by

considering the matrix of ± with respect to a basis of eigenvectors that

χ± (±) = 0.

(v) Show carefully, using (iii) and limiting arguments, that, if ± : C m ’

Cm is linear, then

χ± (±) = 0.

[This argument is substantially longer than that used in algebra courses and

is repugnant to the soul of any pure minded algebraist. But it has its points

of interest.]

Exercise K.285. [13.1, T, ‘] We continue with the ideas of Exercise K.283.

(i) If ±, β ∈ L(U, U ) and ± and β commute, show that

ρ(± + β) ¤ ρ(±) + ρ(β).

(Compare Exercise K.283 (iv).)

(ii) By using the ideas of Exercise K.284, or otherwise, show that, if

± : Cm ’ Cm is linear and > 0, we can ¬nd a β : Cm ’ Cm such that

β < , ± and β commute and ± + β has m distinct eigenvalues.

(iii) By using (ii), or otherwise, show that, if ± : Cm ’ Cm is linear, then

ρ(±) = max{|»| : » an eigenvalue of ±}.

[You may wish to read or reread Exercises K.99 to K.101 at this point.]

Exercise K.286. [13.1, T, ‘] In previous questions we have shown that, if

± : Cm ’ Cm is linear, and we write

ρ(±) = max{|»| : » an eigenvalue of ±}.

then ±n 1/n ’ ρ(±).

Suppose that b ∈ Cm , we choose an arbitrary x0 ∈ Cm and we consider

the sequence

xn+1 = b + ±xn .

590 A COMPANION TO ANALYSIS

If ρ(±) < 1, give two proofs that

xn ’ (ι ’ ±)’1 b ’ 0 as n ’ ∞.

(i) By ¬nding xn explicitly and using Exercise K.282.

(ii) By using the ¬rst paragraph of Exercise K.261.

By considering an eigenvector corresponding to an eigenvalue of largest

modulus show that the sequence will diverge if ρ(±) > 1 and we choose x0

suitably. Show that, if ρ(±) = 1, either we can ¬nd an x0 such the sequence

diverges or, if the sequence always converges, we can ¬nd two starting points

with di¬erent limits.

Exercise K.287. [13.1, T, ‘ ] Consider the problem of solving the equation

Ax = b

where A is an m — m matrix and x and b are column vectors of length

m. If m is small, then standard computational methods will work and, if

m is large and A is a general matrix we have no choice but to use standard

methods. These involve storing all m2 coe¬cients and, in the case of Gaussian

elimination require of the order of m3 operations.

Suppose we have to deal with a matrix A such that A is close to I, in

some sense to be determined later in the question, and there are only of

the order of n non-zero coe¬cients in I ’ A in a well organized pattern.

(Such problems arise in the numerical solution of important partial di¬eren-

tial equations.) The following method can then be employed. Choose x0 and

de¬ne a sequence

xn+1 = b + (I ’ A)xn .

Using the ideas of earlier exercises show that, under certain conditions, to

be stated, xn will tend to a unique solution of Ax = b. Discuss the rapidity

of convergence, and show that, under certain conditions to be stated, only a

few iterations will be required to get the answer to any reasonable degree of

accuracy. Since each iteration requires, at worst, of the order of m2 opera-

tions, and in many cases only of the order of m operations, this method is

much more e¬cient.

The rest of the question consists of elaboration of this idea. We require

Exercise K.286. Suppose that A is an m — m matrix and A = D ’ U ’ L

where L is strictly lower triangular, U is strictly upper triangular and D is

diagonal with all diagonal terms non-zero. We seek to solve Ax = b. The

following iterative schemes have been proposed

Jacobi xn+1 = D’1 (b + (U + L)xn ),

Gauss-Seidel xn+1 = (D ’ L)’1 (b + U xn ).

591

Please send corrections however trivial to twk@dpmms.cam.ac.uk

For each of these two schemes give a necessary and su¬cient condition in

terms of the spectral radius of an appropriate matrix for the method to

work.

Another iterative scheme uses

xn+1 = (D ’ ωL)’1 (ωb + ((1 ’ ω)D + ωU )xn ),

where ω is some ¬xed real number. Give a necessary and su¬cient condition,

in the form ρ(H) < 1 where H is an appropriate matrix, for the method to

work. By showing that

det H = (1 ’ ω)m ,

or otherwise, show that the scheme must fail if ω < 0 or ω > 2.

Exercise K.288. [13.1, S, ‘‘ ] (This is a short question, but requires part (iv)

of Exercise K.281.) Show that Theorem 13.1.13 can be strengthened by

adding the following sentence at the end. ˜Moreover Df |’1 is continuous on

B

B.™

Exercise K.289. [13.1, T, ‘‘ ] This is another exercise in the ideas of Ex-

ercise K.281. We work in (U, U ) as before.

(i) Show that, if ± ∈ L(U, U ), we can ¬nd exp ± ∈ L(U, U ) such that

n

±r

’ exp ± ’ 0

r!

r=0

as n ’ ∞.

(ii) Show carefully that, if ± and β commute,

exp ± exp β = exp(± + β).

(iii) Show that if ± and β are general (not necessarily commuting) ele-

ments of L(U, U ), then

h’2 (exp(h±) exp(hβ) ’ exp(hβ) exp(h±)) ’ (±β ’ β±) ’ 0

as the real number h ’ 0.

Conclude that, in general, exp(±) exp(β) and exp(β) exp(±) need not be

equal. Deduce also that exp(± + β) and exp(±) exp(β) need not be equal.

(iv) Show carefully (you must bear part (iii) in mind) that exp : L(U, U ) ’

L(U, U ) is everywhere continuous.

592 A COMPANION TO ANALYSIS

Exercise K.290. [13.1, P, ‘ ] (i) Consider the map ˜3 : L(U, U ) ’ L(U, U )

given by ˜3 (±) = ±3 . Show that ˜ is everywhere di¬erentiable with

D˜3 (±)β = β±2 + ±β±2 + ±2 β.

(ii) State and prove the appropriate generalisation to the map ± ’ ± m

with m a positive integer.

(iii) Show that exp, de¬ned in Exercise K.289, is everywhere di¬erentiable.

[This requires care.]

Exercise K.291. [13.1, P] Suppose U is a ¬nite dimensional vector space

over C. Let ± : U ’ U be a linear map. If ± has matrix representation A

with respect to some basis, explain why, as N ’ ∞, the entries of the matrix

N n

n=0 A /n! converge to the entries of a matrix exp A which represents exp ±

with respect to the given basis. (See Exercise K.289.)

It is a theorem that any ± ∈ L(U, U ) has an upper triangular matrix with

respect to some basis. By using this fact, or otherwise, show that

det(exp ±) = eTrace ± .

Exercise K.292. [13.1, P] We work in the space M2 (R) of 2 — 2 real ma-

trices. We give M2 (R) the associated operator norm.

(i) Show that the map S : M2 (R) ’ M2 (R) given by S(A) = A2 is

everywhere di¬erentiable with DS(A)B = AB + BA. (If you have done

Exercise K.290, you may just quote it.)

(ii) Show that the matrix equation

01

A2 =

00

has no solution.

(iii) Calculate explicitly all the solutions of

2

ab

= I.

cd

Describe geometrically the linear maps associated with the matrices A such

that A2 = I and det A = ’1. Describe geometrically the linear maps associ-

ated with the matrices A such that A2 = I and det A = 1. Describe geomet-

rically the linear maps associated with the matrices A such that A2 = I and

A is diagonal.

(iv) Show that there are open sets U and V containing 0 (the zero matrix)

such that the equation

A2 = I + X

593

Please send corrections however trivial to twk@dpmms.cam.ac.uk

has exactly one solution of the form A = I + Y with Y ∈ V for each X ∈ U .

(v) Show that we can not ¬nd open sets U and V containing 0 (the zero

matrix) such that the equation

A2 = I + X

has exactly one solution of the form

10

A= +Y

0 ’1

with Y ∈ V for each X ∈ U . Identify which hypothesis of the inverse function

theorem (Theorem 13.1.13) fails to hold and show, by direct calculation, that

is does indeed fail.

(vi) For which B is it true that B 2 = I and we can ¬nd open sets U and

V containing 0 (the zero matrix) such that the equation

A2 = I + X

has exactly one solution of the form A = B + Y with Y ∈ V for each X ∈ U .

Give reasons for your answer.

Exercise K.293. [13.1, P, G] This question requires some knowledge of

eigenvectors of symmetric linear maps. We work in R3 with the usual inner

product. Suppose ± : R3 ’ R3 is an antisymmetric linear map (that is

±T = ’±). Show that x and ±x are orthogonal and that the eigenvalues of

±2 must be non-positive real numbers. By considering eigenvectors of ± and

±2 show that we can always ¬nd µ ∈ R and three orthonormal vectors e1 , e2

and e3 , such that

±e1 = µe2 , ±e2 = ’µe1 , ±e3 = 0.

By choosing appropriate axes and using matrix representations show that

exp ± is a rotation.

Exercise K.294. [13.1, P, G] Show that every linear map ± : Rm ’ Rm

is the sum of a symmetric and an antisymmetric linear map. Suppose ± is

an orthogonal map (that is, ±±T = ι) with ι ’ ± < with very small.

Show that

2

±=ι+ β+ γ

with β , γ ¤ 2.

594 A COMPANION TO ANALYSIS

Exercise K.295. [13.3, M] (Treat this as a ˜methods question™.) The four

vertices A, B, C, D of a quadrilateral lie in anti-clockwise order on a circle

radius a and center O. We write 2θ1 = ∠AOB, 2θ2 = ∠BOC, 2θ3 = ∠COD,

2θ4 = ∠DOA. Find the area of the quadrilateral and state the relation that

θ1 , θ2 , θ3 and θ4 must satisfy.

Use Lagrange™s method to ¬nd the form of a quadrilateral of greatest area

inscribed in a circle of radius a. (Treat this as a ˜methods question™.)

Use Lagrange™s method to ¬nd the form of an n-gon of greatest area

inscribed in a circle [n ≥ 3].

Use Lagrange™s method to ¬nd the form of an n-gon of least area circum-

scribing a circle [n ≥ 3].

[Compare Exercise K.40.]

Exercise K.296. [13.3, T] Let p and q be strictly positive real numbers

with p’1 + q ’1 = 1. Suppose that y1 , y2 , . . . , yn , c > 0. Explain why there

must exist x1 , x2 , . . . , xn ≥ 0 with n xp = c and

j=1 j

n n n

tp = c.

xj yj ≥ tj yj whenever t1 , t2 , . . . , tn ≥ 0 with j

j=1 j=1 j=1

Use the Lagrange multiplier method to ¬nd the xj . Deduce from your

answer that

1/p 1/q

n n n

|aj |p |bj |q

|aj bj | ¤

j=1 j=1 j=1

whenever aj , bj ∈ C. Under what conditions does equality hold?

This gives an alternative proof of the ¬rst result in Exercise K.191 (i).

Exercise K.297. (The parallelogram law.) [14.1, T] Except in the last

part of this question we work in a real normed vector space (V, ).

(i) Suppose that V has a real inner product , such that x, x = x 2

for all x ∈ V . Show that

2 2 2 2

+ x’y

x+y =2 x +2 y

for all x, y ∈ V . (This is called the parallelogram law.)

(ii) Show also that

2 2

’ x’y

4 x, y = x + y

for all x, y ∈ V . (This is called the polarisation identity.)

595

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) Use the parallelogram law to obtain a relation between the lengths

of the sides and the diagonals of a parallelogram in Euclidean space.

(iv) Prove the inequality | x ’ y | ¤ x ’ y and use it together with

the polarisation identity and the parallelogram law to give another proof of

the Cauchy-Schwarz inequality.

(v) Suppose now that (V, , ) is a complex inner product space with

norm derived from the inner product. Show that the parallelogram law

holds in the same form as before and obtain the new polarisation identity

2 2 2

’ i x ’ iy 2 .

’ x’y

4 x, y = x + y + i x + iy

(vi) Show that the uniform norm on C([0, 1]) is not derived from an inner

product. (That is to say, there dooes not exist an inner product , with

f 2 = langlef, f for all f ∈ C([0, 1]).

∞

Exercise K.298. [14.1, T, ‘ ] The parallelogram law of Question K.297

actually characterises norms derived from an inner product although the

proof is slightly trickier than one might expect.

(i) Let (V, ) be real normed space such that

2 2 2 2

+ x’y

x+y =2 x +2 y

for all x, y ∈ V . It is natural to try setting

x, y = 4’1 2 2

’ x’y

x+y .

Show that x, y = y, x for all x, y ∈ V and that x, x = x 2 (so that,

automatically, x, x ≥ 0 with equality if and only if x = 0).

(ii) The remaining inner product rules are harder to prove. Show that

2 2 2 2

+ u+v’w

u+v+w =2 u+v +2 w

and use this to establish that

u + w, v + u ’ w, v = 2 u, v (1)

for all u, v, w ∈ V . Use equation (1) to establish that

2u, v = 2 u, v (2)

and then use equations (1) and (2) to show that

x, v + y, v = x + y, v

596 A COMPANION TO ANALYSIS

for all x, y, v ∈ V .

(iii) Establish the equation

»x, y = » x, y

for all positive integer values of », then for all integer values, for all rational

values and then for all real values of ».

(iv) Use the fact that the parallelogram law characterises norms derived

from an inner product to give an alternative proof of Lemma 14.1.11 in the

real case.

(v) Extend the results of this question to complex vector spaces.

Exercise K.299. [14.1, P] Suppose (X, d) is a complete metric space with

a dense subset E. Suppose that E is a vector space (over F where F = R

or F = C) with norm E such that d(x, y) = x ’E y E for all x, y ∈

E. Suppose further that there is map ME : E 2 ’ E such that, writing

ME (x, y) = xy, we have

(i) x(yz) = (xy)z,

(ii) (x + y)z = xz + yz, z(x + y) = zx + zy

(iii) (»x)y = »(xy), x(»y) = »(xy),

(iv) xy ¤ x y ,