Exercises for linear algebra - Dynamics-approx.jku.at

(Use the fact that B = A t A). (We remark that matrices of the type B arisenaturally in the discretisation of Sturm-Liouville operators).1.3 General exercisesWe conclude with some exercises on general properties of matricesExample Let A be an invertible n×n matrix such that the row sums of Aare constant. Show that this constant is non-zero and that the inverse of Aalso has non constant row sums. What is the common value of these sums?Example Show that if an m × n matrix A is such that A t A = 0, thenA = 0.Example Show that every invertible n×n matrix is a product of matricesof one of the following three forms⎡ ⎤0 0 0 ... 11 0 0 ... 0⎢. .,⎥⎣ 0 ... 1 0 0 ⎦0 ... 0 1 0⎡ ⎤0 1 0 ... 01 0 0 ... 00 0 1 ... 0⎢ ⎥⎣ . . ⎦0 0 0 ... 1andand⎡ ⎤1 k 0 ... 00 1 0 ... 0⎢ ⎥⎣ . . ⎦ ,0 0 0 ... 1⎡⎢⎣λ 0 0 ... 00 1 0 ... 0. .0 0 0 ... 16⎤⎥⎦

(Note that the last three are elementary matrices i.e. correspond to elementaryrow operations).Example Recall that a square matrix A is doubly stochastic if its elementsare non-negative and the sum of each of its row and columns is 1.Show that the latter condition is equivalent to the fact that Ae = e = A t ewhere e is the column matrix with all entries “1”s. If y = (η 1 ,...η n ) andx = (ξ 1 ,...,ξ n ), we say that y dominates x ifandξ 1 ≤ η 1 ,ξ 1 +ξ 2 ≤ η 1 +η 2 ,...,ξ 1 +···+ξ n−1 ≤ η 1 +···+η n−1ξ 1 +···+ξ n = η 1 +...η n .Show that this is equivalent to the fact that there is a doubly stochasticmatrix A so that X = AY. (X and Y are the column matrices correspondingto x and y).Example Let A be an m×n matrix, B and n×p one. Show thatr(AB) ≥ r(A)+r(B)−n.Deduce that if r(A) = n, r(AB) = r(B).Example Let M be a subset of the set M n of n × n matrices with theproperties• if A,B ∈ M,λ,µ ∈ R, then λA+µB ∈ M;• if P,Q ∈ M n , A ∈ M, then PAQ ∈ M.Show that M is either {0} or M n .2 Geometry2.1 The plane as a vector spaceWe begin with a number of problems which illustrate the use of the vectorspace structure of R 2 to obtain geometrical results:II.1.A Calculate the barycentric coordinates of the point p with respect toA, B and C where the numbers in Figure 1 indicate the ratios in which thevarious lines are divided.7

II.1.B Show that the in-centre of the triangle ABC is the point1s (ax A +bx B +cx C )where a−|BC|, b = |CA|, c = |AB|, s = a+b+c. What is the correspondingresult for the out-centres?II.1.C What are the barycentric coordinates of P,Q and R with respect toA, B and C? Use this to whose that the area of ABC is seven times that ofPQR. (See Figure 2).II.1.D Whatistheareaoftheconvexn-gonwithverticesP 1 = (ξ 1 1 ,ξ1 2 ),...,P n =(ξ n 1,ξ n 2)?II.1.E Let X and Y be the midpoints of AC resp. BD and W the intersectionof AD and BC. Show that the area of WXY is one quarter of thatof the rectangle ABCD (Figure 3).II.1.F Show that the midpoints of the sides of a quadrilateral form thevertices ofaparallelogram. What istheratiooftheareaofthisparallelogramto that of the originalquadrilateral? (Figure 4).II.1.G Show that if M is the centroid of the triangle ABC, then the areaof AC 1 M is a sixth of that of ABC where C 1 is the midpoint of AB (Figure5).II.1.H Let A 1 , A 2 , A 3 , A 4 be the vertices of a square and let B 1 , B 2 ¡ B 3and B 4 be as in diagram 6. Show that they also form the vertices of a squareand that the area of the latter is one fifth of that of the original one.II.1.I Show that a quadrilateral ABCD is a parallelogram if and only ifAC and BD cross at their midpoints. Show that if ABCD and APCQ areparallelograms (see Figure 7), then BPDQ is also a parallelogram (whichmay be degenerate i.e. such that its vertices are collinear).II.1.J Let ABCD be a quadrilateral and choose B 1 , D 1 as in diagram 8 sothat CB 1 and AB (resp. CD 1 and AD) are equal and parallel. Show that• BB 1 DD 1 is a parallelogram;8

• the lines BC, DC, D 1 C and B 1 C are equal and parallel to sides of theoriginal quadrilateral and the angles between them are the angles ofthe original quadrilateral;• the area of BB 1 D 1 D is twice that of ABCD.(see Figure 9).II.1.K Let ABC be a triangle and let the bisectors of the angles at A resp.B meet the opposite sides at A 1 and B 1 . Show that|BA 1 |/|CA 1 | = |AB|/|AC|and that if |CC 1 | = |BB 1 |, then |AB| = |AC|. Show that ifthen |BA|/|AC| = λ1−λ .x A1 = x B +(1−λ)x CII.1.L Show that the bisectors of the angles between the lines(x−x 0 |n 0 ) = 0 and (x−x 1 |n 1 ) = 0have equations(x−x 0 |n 1 ) = ±(x−x 1 |n 1 ).II.1.M ProvethefollowinggeneralisationofMenelaus’theorem: A 1 ,...,A 5are the vertices of a pentagon and a line L meets the edges in the pointsP 1 ,...,P 5 as in figure ?? Show that|P 1 A 1 |· |P 2A 2 |· |P 3A 3 |· |P 4A 4 |· |P 5A 5 |= 1.|P 1 A 1 |P 2 1A 3 |P 3 A 4 |P 4 A 5 |P 5 A 6Can you find a similar generalisation of Ceva’s theorem?II.1.N Let tL be the line{x : (n|x) = (n|x 0 )where n is the unit normal. Show that the distance from y to L is |(n|y−x 0 ).9

II.1.O Let tL 1 , L 2 and L 3 through a fixed point P and let A 1 and B 1 (resp.A 2 and B 2 resp. A 3 and B 3 ) be points on L 1 resp. L 2 resp. L 3 . Show that ifA 1 A 2 is parallel to B 1 B 2 and A 2 A 3 is parallel to B 2 B 3 , then A 3 A 1 is parallelto B 3 B 1 .II.1.P Let A, B, C and D be non-collinear points in the plane such that|AB| = |CD| and |AD| = |BC|. Show that x AD − x BC ⊥ x AC and x Ad −x BC ⊥ x BD and that x AD = x BC or AC is parallel to BD.II.1.Q Let L = L a,b,c and L 1 = L a1 ,b 1 ,c 1be anon-parallel lines and putL λ,µ = {(ξ 1 ,ξ 2 ) : λ(aξ 1 +bξ 2 +c)+µ(a 1 ξ 1 +b 1 ξ 2 +c 1 ) = 0}.Show that {L λ,µ ;λ,µ ∈ R,λ+µ = 1} is the set of all lines through the pointof intersection of L 1 and L 2 .II.1.R Let x, y and z be points in the plane. Show that x and y areparallel if and only if we have ‖x+y‖ = ‖x‖+‖y‖ or ‖x−y‖ = ‖x‖+‖y‖andthat bothof these conditionsareequivalent tothevalidity ofthe equality|(x|y)| = ‖x‖‖y‖. Show that z is the midpoint of x and y if and only if wehave the equalities‖x−z‖ = ‖z −y‖ = 1 2 ‖x−y‖.II.1.S Let ABC be a triangle and P, Q and S as in the diagram. Showthat the area of the triangle PQR is one seventh that of ABC.II.1.T Let x 1 , x 2 , x 3 and x 4 be vectors in the p lane. Show that thereexists a triangle ABC with x AB = x 1 , x BC = x 2 and x CA = x 3 if and only ifx 1 +x 2 +x 3 = 0. Show that there is a parallelogram ABCD with x AB = x 1etc. if and only if x 1 +x 2 +x 3 +x 4 = 0 and x 1 +x 3 = 0.II.1.U Let A, B, C and D be points in the plane and putx A ′ = 1 2 (x B +x C ) x B ′ = 1 2 (x C +x A ) x C ′ = 1 2 (x A +x B ).Show that(x A ′ D|x BC )+(x B ′ D|x CA )+(x C ′ D|x AB ) = 0.10

II.1.V Let a, b and c be vectors in the plane. Does there always exist atriangle ABC so that x A,1(B+C) = a etc. How many such triangles exist?2II.1.W Let x, y, z be points in the plane. Show that‖x+y −z‖ 2 = ‖x−z‖ 2 +‖y −z‖ 2 −‖x−y‖ 2 +‖x‖ 2 +‖y‖ 2 −‖z‖ 2 .2.2 Affine mappings of the planeThe essential reason for the success of applying methods of linear algebra togeometry is the fact that the interesting elementary transformations of theplane are affine and so essentially implemented by matrices. We bring someexercises on this them.II.2.A Calculate the images resp. the pre-images of the lineaξ 1 +bξ 2 +c = 0and the conic sectionunder the affine mappingaξ 2 1 +2bξ 1ξ 2 +cξ 2 2 = 0(where a 11 a 22 −a 12 a 21 ≠ 0).(ξ 1 ,ξ 2 ) ↦→ (a 11 ξ 1 +a 12 ξ 2 +c 1 ,a 21 ξ 1 +a 22 ξ 2 +c 2 )II.2.B More generally, find the image and pre-image of the conic section(f(x)|x)+2(b|x)+c = 0(where f is an affine mapping with symmetric matrix and b is a vector inR 2 ) under the affine mapping x ↦→ g(x)+u (where g is linear and injective).II.2.C Show that if f is an injective affine mapping and x,y,z are collinearand distinct, then‖f(x)−f(y)‖‖x−y‖= ‖f(x)−f(z)‖ .‖x−z‖II.2.D Let f be a linear, invertible mapping on R 2 . Show that there is anorthonormal basis (x 1 ,x 2 ) for R 2 so that f(x 1 ) ⊥ f(x 2 ). for which f is thereprecisely one such basis?11

2.3 CirclesII.3.A Show that the lineaξ 1 +bξ 2 +c = 0cuts the circleif and only if‖x−x 0 ‖ 2 = r 2(ξ 0 1 +bξ 0 2 +c) 2 ≤ r 2 (a 2 +b 2 ).II.3.B consider the circlesξ 2 1 +ξ 2 2 +2aξ 1 +2bξ 2 +c = 0 (1)ξ 2 1 +ξ 2 2 +2aξ 1 +2bξ 2 +c 1 = 0 (2)with radii r resp. r 1 . Put d 2 = (a−a 1 ) 2 +(b−b 1 ) 2 . For which values of r,r 1and d do the circles intersect? If they do intersect, show that the angle θ atwhich they cross each other is given by the formulacosθ = r2 +r 2 1 −d 22rr 1.Deduce a criterium for the circles to cut at right angles.II.3.C If C is the circle ‖x−a‖ 2 = r 2 in R 2 , the power of a point x withrespect to C is the pointp(x) =Interpret this geometrically.II.3.D Show that the line{x 0 +tu : t ∈ R}(where u is a unit vector) cuts the circle ‖x = a‖ 2 = r 2 at the points correspondingto the roots of the quadratic equationin t. Show that the product of these roots is independent of u. Interpret thisresult geometrically (bear in mind the case where x 0 lies inside of the circle).12

II.3.E Denote by C a,b,c the circleξ 2 1 +ξ 2 2 −2aξ 1 −2bξ 2 −c = 0.• show that if x = (ξ 1 ,ξ 2 ) is a point in R 2 , thenS C (x) = ξ 2 1 +ξ2 2 −2aξ 1 −2bξ 2 +cis the square of the length of the tangent from x to C = C a,b,c providedthat x is exterior to C;• if C 1 = C a1 ,b 1 ,c 1, then C and C 1 are tangential to each other if and onlyif2aa 1 +2bb 1 −(c+c 1 +2rr 1 ) = 0where r resp. r−1 are the radii of the circles;• if C and C 1 are circles as above, then{x ∈ R 2 : S C (x) = S C1 (x)}is a straight line. It is called the radial axis of C and C 1 (give ageometrical interpretation);• show that if C, C 1 and C 2 are circles so thatthen the radial axes of the three circles are concurrent;• show that if λ ≠ 1, thenC λ = {x ∈ R 2 : S C (x) = λS C1 (x)}isacircleandthatifC andC 1 intersect intwopoints, thenitrepresents(with varying λ) the family of circles passing through these points;• show that if a circle C 2 meets C and C 1 at right angles, then it alsomeets each C λ at right angles (C, C 1 and C λ as in (5)).II.3.F Show that if C 1 , C 2 and C 3 are three circles, no two of which areconcentric, then the radial axes (cf. E above) are concurrent or parallel (thepoint of intersection in the former case is called the power point of thethree circles).13

II.3.G Show that if A 1 , A 2 , A 3 , A 4 , A 5 and A 6 are six points on a circle sothat the linesA 1 A 4 ,A 2 A 5 ,A 3 A 6meet in a point, then|A 1 A 2 |·|A 3 A 4 |·|A 5 A 6 | = |A 2 A 3 |·|A 4 A 5 |·|A 6 A 1 |.Can you generalise this result to 2n+2 points on a circle (whereby n > 1)?II.3.G Show that if A 1 ,...,A 2n+1 are the vertices of a regular polygoninscribed in a circle of unit radius and L i is the length of A 1 A i+1 , thenn∑ n∏L 2 i = L i .i=1II.3.H Let ABC be a triangle, A 1 a point on BC, B 1 on CA, C 1 on AB.Show that the circles through AB 1 C 1 , BC 1 A 1 and CA 1 B 1 are concurrent.2.4 Conic sectionsThe next most complicated class of curves after the circles are the conicsections which were also studied by the Greeks. Their classification is one ofthe highpoints of plane geometry. Similar methods can be used to prove anumber of elegant results on conic sections.II.4.A Show that if f : R 2 → R 2 is an injective, affine mapping, then fmaps conic sections onto conic sections. For which f is the image of a circlealso a circle?i=1II.4.B Show that if Q is the conic section{x ∈ R 2 : (f(x)|x)+2(b|x) = c = 0}then the tangent to Q at x 0 is the line{x ∈ R 2 : (f(x)|x 0 )+(b|x+x 0 )+c = 0}.What is the geometrical significance of this line if x 0 does not lie on Q?14

• the conic has the form{x : X t DX = 0}whereX =[ξ1ξ 2];• the image of Q under the linear mapping with matrix U has the correspondingmatrix [ ] U t AU U t BB t U ?;• that the following numerical functions of the conic section are invariantunder such a transformation:trA, d 0 = detA, d = detD;• that Q is central (or empty) if and only if d 0 ≠ 0 and d ≠ 0;• if d 0 > 0, then Q is an ellipse or empty;• if d 0 < 0, then Q is a hyperbola;• if d 0 = 0 and d ≠ 0, then Q is a parabola;• if d = 0 and d 0 > 0, then Q is a point;• if d = 0, d 0 < 0, then Q is a pair of intersecting lines;• if d = d 0 = 0, then Q is a line, a pair of parallel lines or empty.II.4.G Calculate the locus of the foci of all conics which touch the foursides of a given parallelogram.II.4.H Let A and B be fixed points in the plane and suppose that a pointC moves on a fixed circle with centre A. Find the locus of the point P whereP is the intersection of BC and the internal bisector of the angle ABC.16

3 IsometriesOneofthefundamental concepts ofeuclidean geometryisthatofcongruence.This corresponds to the modern notion of isometry. The basic result on theisometries of R 2 is their classification into the following types• translations• reflections• rotations• glide reflections.3.1 Concrete isometriesIII.1.A Show that if f is a half-turn about the point P, then P is themidpoint of x and f(x) for any x in R 2 .III.1.B Show that if P, Q and R are points in the plane, then the relationshipx Q,π ◦D xP ,π = T xPr = D xR ,π ◦D xQ ,πholds if and only if fA is the midpoint of PR.III.a.C Show that if f and g are isometries, then G −1 ◦f ◦g is an isometryof the same geometrical type as f.III.1.D Show that a glide reflection of the plane can be expressed as aproduct of three reflections in the sides of a triangle. (See figure ??).III.1.E Show that a mapping of the formD xP ,π ◦D xQ ,π ◦D xP ,πis a half-turn and calculate its axis. Carry out a similar analysis of theisometriesD xP ,π ◦T u ◦D xP ,π T u ◦D xP ,π ◦T −u .III.1.F Show that if L 1 , L 2 and L 3 are non-concurrent lines, thenis a translation.(R L1 ◦R L2 ◦R L3 ) 217

III.1.G Which isometries of the plane satisfy the condition f 2 = Id?III.1.H Let L 1 and L 2 be distinct one-dimensional subspaces of R 2 andlet x be a point on the bisector of the angle between them. Describe thesuccessive images of x under the mappingsR L1 , R L2 ◦R L1 , R L1 ◦R L2 ◦R L1 ,....Distinguish between the cases where the angle between the lines has the form2πα where• α is 1 n(n ∈ N)• α is rational• α is irrational.III.1.I Characterise those matrices[ ]a11 a 12a 21 a 22for which the corresponding linear mappinghas the property thatwhere (x|y) R = ξ 1 η 1 −ξ 2 η 2 .3.2 Pseudo-squaresf : (ξ 1 ,ξ 2 ) ↦→ (a 11 ξ 1 +a 12 ξ 2 ,a 21 ξ 1 +a 22 ξ 2 )(f(x)|f(y)) R = (x|y) RA pseudo-square is a rectangle ABCD for which AC and BD are equal inlength and perpendicular to each other. The use of the rotation operatorallows an elegant approach to their theory.III.2.A Show that ABCD is a pseudo-square if and only ifx AC = D ±π2 (x BD).III.2.B Show that the definition is equivalent to the existence of an F sothat AFD and BFC are right-angled, isosceles triangles. Show that therethen exists a G so that BGA and CGD are also right-angled and isosceles.18

III.2.C Show that if M 1 (resp. M 2 ) is the midpoint of BD (resp. AC,then CGM 1 M 2 is a square with centre the centroid of A,B,C and D (i.e. thepoint 1 4 (x A +x B +x C +x D ).III.2.D Show that F and G lie on the bisectors of the angles between ACand BD.III.2.E Show that|AD| 2 +|BC| 2 = |AB| 2 +|CD| 2and that the angles ∠FAD and ∠GAB are equal.III.2.F ShowthatthemidpointsofAB, BC, CD andDAformtheverticesof a square.III.2.G Show that if λ is a fixed real number and we definex P = λx A +(1−λ)x B x Q = λx B +(1−λ)x C (3)x R = λx C +(1−λ)x D x S = λx B +(1−λ)x A (4)then PQRS is also a pseudo-square.III.2.H Show that if A 1 B 1 C 1 D 1 is a second pseudo-square and λ is a fixedreal number, then A 2 B 2 C 2 D 2 is also a pseudo-square, wherex A2 = λx A +(1−λ)x A1 etc.3.3 ConstructionsAnother use of isometries is to provide elegant solutions ofconstruction problemsof the following type.III.3.A GiventwofixedlinesL 1 andL 2 andapointAERasinthediagram,show how to find points B, C and D as in figure ?? with B on L 1 and D onL 2 so that ABCD is a square.III.3.B L, A, and B are given as in figure ?? Show how to construct apoint P on L so that LAP = LAB.19

III.3.C Given a circle and a point A outside of it, show how to constructa line through A which meets the circle in points P and Q so that P is themidpoint of AQ (figure ??).III.3.D Given a triangle ABC and a segment p as in diagram ??, showhow to find a line L, parallel to p, which meets the triangle at the endpointsof a segment which is equal to p.III.3.E Given two circles C 1 and C 2 and a line L, find a line L 1 which isparallel to L and is such that the segment AB has a given length a where Aand B are as in diagram ??III.3.F Given a circle C, a line L and a point A, find a line L 1 through Aso that |PA| = |PB|. (see figure ??III.3.G Given intersecting circles C 1 and C 2 find a chord through a pointof intersection A so that the difference |PA|−|AQ| is equal to a given lengtha.III.3.H Given a line L and curves C 1 and C 2 on opposite sides of L, constructa perpendicular to L which meets C 1 and C 2 at points equidistant toL (figure ??III.3.I GivenalineLandcirclesC 1 andC 2 onoppositesidesofL, constructa square with two vertices on L and one each on C 1 and C 2 .3.4 Some geometrical resultsIn this section, we collect some attractive geometric results which can beproved elegantly by using isometries.III.4.A Let ABCD be a non-degenerate convex quadrilateral and denoteby X the centre of the external square on AB. Y, Z and W are constructedsimilarlywithrespect toBC, CD andDA(seefigure??). ShowthatXYZWis a pseudo-square (i.e. XZ and YW areequal andperpendicular—cf. III.4).III.4.B Consider diagram ?? where AC ′ B and AB ′ C are similar isoscelestriangles. Show that AB ′ A ′′ C and AB ′′ A ′ C ′′ are parallelograms.20

III.4.C Let ABCD be a quadrilateral and P, Q, R and S be so that thetriangles PDC, ARD AQB and BSC are equilateral (figure ??). Show thatPQ and RS are equal and parallel (i.e. that ) What can you say aboutPQRS in the special case where ABCD is a parallelogram?III.4.D Consider diagram ?? where ABC is a triangle and P, Q and Rare the centres of the external triangles which are equilateral. Show thatPQR is equilateral (This is known as Napoleon’s theorem and is sometimesattributed to him).III.4.E Let A, B, C be a triangle, M the midpoint of the side BC. Showthat if Z 1 and Z 2 are the centres of the exterior squares on the sides AB andBC resp., then Z 1 MZ 2 is an isosceles, right-angled triangle (figure ??).III.4.F Let A 1 , A 2 , A 3 and A 4 be points on a circle C and denote by H 1 ,H 2 , H 3 and H 4 the orthocentres of the triangles, A 2 A 3 A 4 , A 3 A 4 A 1 , A 4 A 1 A 2and A 1 A 2 A 3 . Show that there is a point O so that H i = D xO , π 2 (A i) for eachi.III.4.G Let ABCD be a quadrilateral, B 1 and D 1 as in figure ?? Show• that BB 1 DD 1 is a parallelogram;• that the sides of ABCD are equal and parallel to the segments from Cto the vertices of BB 1 D 1 D (under a suitable pairing);• that the angles described at C by these lines are the same as the anglesof the quadrilateral ABCD;• that the area of BB 1 D 1 D is twice that of ABCD.3.5 Isometries in R 3Once again, one can, with the aid of the methods of linear algebra, cataloguethe possible isometries of R 3 . They are• translations;• rotations;• rotary translations;• glide-reflections;21

• rotary reflections;• screw displacements.III.5.A Describethegeometricformofthesetsofsuccessiveimages{f n (x) :n ∈ N} of a point x in R 3 under each of the above types of isometry.III.5.B Let f be an isometry of R 3 . Show that if x ∈ R 3 , thenf(x)×f(y) = (detf)(x×y)for each x,y ∈ R 3 . Hence we have the equation f(x)×f(y) = x×y if f isa proper motion (i.e. its determinant is positive and so has the value 1).III.5.C A rigid motion of R 3 is a continuous mapping T from [0,1] intothe space of isometries of R 3 so that T(0) = Id. (Here continuity means thatthe elements of the matrices of the isometries T(t) depend continuously ont). Show that an isometry f is proper (cf. III.5.B) if and only if there is arigid motion T so that T(1) = f.III.5.D Let U t be the screw displacementT (0,0,t) ◦D [(0,0,1)],t .Calculate the matrix of U t and show that U t+s = U t ◦U s . Calculate the path{U t (1,0,0) : t ∈ R}of the point (1,0,0) under this motion. What are the orthogonal projectionsof this path onto• the (x,y)-plane;• the (y,z)-plane;• the (z,x)-plane;• the plane [(0,1,0),( 1 √2,0,1√2)].III.5.E Let f be a linear mapping on R 3 . Show that there is a rotation gso that f ◦g has a diagonal matrix with respect to the canonical basis.22

3.6 Directed anglesIn this section we return to the plane. Using the scalar product, it is simpleto define the angle between two vector and hence between two intersectinglines. However, in some situation, it is necessary to distinguish between thetwo possible directions (clockwise and anti-clockwise) along which an anglecan be traversed. This requires a slightly more sophisticated approach whichwe now describe.Firstly a definition: Let O and A be points in the plane. The ray OA(written r OA ) is the set{x O +tx OA : t > 0}.III.6.A Show thatr OA = {P : |AP| = |OA|−|OP|}.III.6.B Show that if B ∈ r OA , then r OB = r OA .III.6.C Show that two rays r OA and r OB are either disjoint or equal.III.6.D Show that there is a unique θ ∈ [0,2π[ so thatD xO ,θ(r OA = r OB .θ is then called the directed angle from OA to OB (written ∠(r OA ,r OB )).III.6.E Show that if ABC is a triangle, then4 Solid geometryIV.1.A Show that the lines∠(r AB ,r AC )+∠(r BC ,r BA )+∠(r CA ,r CB ).L a,b,c ,L a1 ,b 1 ,c 1,L a2 ,b 2 ,c 2are concurrent or parallel if and only if the vectors (a,b,c), (a 1 ,b 1 ,c 1 ) and(a 2 ,b 2 ,c 2 ) are linearly dependent in R 3 . Use this to show that the bisectorsof the angles of a triangle are concurrent (do the same for the altitudes andthe perpendicular bisectors of the sides).23

IV.2.A Find an explicit formula for the distance from the point x in spaceto the plane spanned by the three vectors y 1 ,, y 2 and y 3 .IV.2.B Let L 1 and L 2 be skew lines in space (i.e. they ;do not meet andare not parallel). Let P be a point in space which lies on neither of theselines. Show that there is precisely one line through P which meets L 1 andL 2 . If L is L (a,b,c,d) and L 1 is L a1 ,b 1 ,c 1 ,d 1, calculate a formula for the distancebetween the lines.IV.1.D Calculate the area of the projection of the ellipsoid( ) 2 ξ1+a( ) 2 ξ2+bon the plane perpendicular to the unit vectorn = (n 1 ,n 2 ,n 3 ).( ) 2 ξ3= 1cIV.1.E Calculate the eccentricity of the ellipse shown in figure ??IV.1.F Let V be the ellipsoid as in D above. Find an affine mapping f onR 3 which maps V onto the unit ball and use this to calculate its volume.4.1 TetrahedraIV.2.A Let ABCD be a skew quadrilateral in R 3 (i.e. the vertices are notcoplanar). Show that the line joining the midpoints of AB and CD intersectand bisects the line joining the midpoint of AC and BD.IV.2.B Let A, B, C and D be the vertices of a tetrahedron in space. Showthat if AB and CD are perpendicular (resp. AC and BD are perpendicular),then AD and BC are also perpendicular.IV.2.C Show that if one altitude of a tetrahedron intersects two others,then all four are concurrent.IV.2.D Show that the lines joining the vertices of a tetrahedron ABCD tothe centroids of the opposite faces are concurrent and cut each other in theratio 3 : 1.24

IV.2.D Let x, y and z be three vectors in space and denote by H (resp.K) the volume of the parallelotope spanned by them (res. the parallelotopewith these vectors as altitudes). Show thatHK = ‖x‖ 2 ‖y‖ 2 ‖z‖ 2 .IV..2.E Let x, y and z be vectors in space. Show that(x|x)(y|z) ≥ (x|z)(x|y) = ‖x×z‖‖x×y‖.Deduce that if O, A, B and C are points in space as in diagram ?? then∠AOC ≤ ∠AOB +∠BOC.IV.2.F If x, y and z are as above, show that‖x−y‖+‖y −z‖+‖z −x‖ ≥ ‖(x−y)×(z −y)‖.Deduce that if O, A, B and C are as in diagram ??, then4.2 The Platonic bodies△ABC ≤ △OAB +△OBC +△OCA.A Platonic body is a polyhedron which is regular in the sense that its facesconsist ofcongruent regularpolygonsanditsvertices areallsimilar. thereareprecisely five such bodies—the tetrahedron, hexahedron (cube), octahedron,dodecahedron and icosahedron. The next exercises are devoted to some oftheir properties.IV.3.A Consider the points A = (1,0,0), B = (0,1,0) and C = (0,0,1).Then there are two points O 1 and O 2 so thatresp.|) 1 A| = |) 1 B| = |) 1 C| = √ 2|O 2 A| = |O 2 B| = |O 2 C| = √ 2.Chose one of these points and denote it by O. Then OABC is a regulartetrahedron. Calculate• its surface area;• its volume;25

• the angle between adjacent sides;• the orthogonal projection of OABC on the (x,y)-plane resp. the planeorthogonal to (1,1,1).Deduce from (1) and (2) an expression for the volume and surface area of aregular tetrahedron in terms of the length of a side.IV.3.B The eight point A = (1,1,1), B = (−1,1,1), C = (1,−1,1),D = (1,1,−1), F = (1,−1,1), G = (−1,1,−1), H = (−1,−1,1) andE = (−1,−1,−1) form the corners of a cube. Prove• that A, H, F and G are the vertices of a regular tetrahedron;• that the midpoints of the sides AB, BH, HE, EF, FD and DA lieon a plane and form the vertices of a regular hexagon. Calculate thelatter’s area;• that the points A, B, G, D and H are the vertices of a pyramid.Calculate its volume and the angle between adjacent sides. Show thatthe cube can be dissected into three such congruent pyramids;• that if 0 < λ < 1, then the points A 1 , B 1 , C 1 and D 1 form the verticesof a square whereA 1 = (−1,−1+λ,1) B 1 = (1−λ,1,1) C 1 = (1,1−λ,−1) D 1 = (−1+λ,−1,−1).For which λ is the area of this square maximal?IV.3.C The centroids of the faces of the above cube form the verticesof a regular octahedron, as do the midpoints of the sides of the regulartetrahedron. Calculate the surface area and volume (as functions of thelength of a side) and the angle between adjacent sides.IV.3.D If b > 0, then the twelve points(0,±1,±b) (±b,0,±1) (±1,±b,0)form the vertices of an icosahedron. Show that if b = τ (the golden mean12 (1+√ 5) (see ??), then this is a regular icosahedron. Calculate• the length of its sides;• the surface area;26

• the volume;• the angle between adjacent sides.IV.3.E Calculate the centroids of the twenty faces of the regular icosahedronand show that they form the vertices of a regular dodecahedron.Calculate the quantities (1) - (4) of the last exercises for this solid.IV.3.F Calculate the normal to one of the faces of the dodecahedron andthe orthogonal projection of its vertices onto the plane of this face.IV.3.G Prove the formulae√ √R 3( 5+1)a = 4√ra = (25+ √ 152 √ 10for the regular dodecahedron and√ √√√ (Ra = 5+ √ )52 √ 2ra + 3+√ 54 √ 3fortheregularicosahedron, whereRistheradiusofthecircumscribed sphere,r is the radius of the inscribed sphere and a is the length of a side.IV.3.H The rhombic dodecahedron is the polyhedron with vertices(0,0,±2) (0,±2,0) (±2,0,0) (1,1,1) (1,1,−1) (5)(1,−1,1) (−1,1,1) (−1,−1,1) (1,−1,−1) (−1,1,−1). (6)Calculate the angle between two faces, its surface area and the angles of therhombi which constitute its faces.IV.3.I Let A, B, C and D be the points (1,0,0,0), (0,1,0,0), (0,0,1,0)and (0,0,0,1) in R 4 . Show that there are two points E and F so that A, B,C, D and E resp. A, B¡ C, D and F are the corners of a regular polytope inR 4 . Calculate the angle between the sides of this polytope and its projectiononto the x,y,z space resp. the x,y plane resp. on the space{x ∈ R 4 : ξ 1 +ξ 2 +ξ 3 +ξ 3 = 0}.27

IV.3.J Inthis last exercise we consider thehypercube inR 4 . It hasverticesthe points with coordinates (±1,±1,±1,±1). How many edges resp. sidesdoes it have. What is the length of its diagonal?5 Vector spaces and linear mappings5.1 ?V.1.A Show that two affine subspaces T u (M) and T v (M) of a vector spaceV coincide if and only if u ∈ T v (M) or equivalently v −u ∈ M.V..1.B Show that if two affine subspaces of the form T u (M) and T v (N)coincide, then M = N.V.1.C Show that the affine subspace spanned by the set{x 1 ,...,x n }consistsofthosevectorsoftheform ∑ ni=1 λ ix i where ∑ ni=1 λ i =1.Show that this representation is unique if and only if the x i are affinelyindependent.5.2 ?V.2.A If u is a vector in R 3 , then the mappingx ↦→ u×xis linear. What is its matrix with respect to the canonical basis?V.2.B Suppose that M is a subspace of L(V) with the property that theonly subspaces of V which are invariant under each f ∈ M are {0} and Vitself. Show that if g ∈ L(V) commutes with each element of M, then g iseither the zero operator or invertible.6 Determinants6.1 The calculation of concrete determinantsThe literature abounds with exercises involving the explicit calculation ofdeterminants, starting with the Vandermonde determinant. We bring a selectionof some of the more interesting ones:28

VI.1.A Calculate the determinant of the following n×n matrices A = [a ij ],whereby• a ij −g.c.d.(i,j);• a ii = 2cosθ, a i,i−1 = a i,i+1 = 1, a ik = 0 otherwise;• a ij = ad ij −b i b k ;• a ij =1(i+j +1)!(0 ≤ i,j ≤ n−1);• the a ij are given recursively by the formulae:a 11 = 1,a i1 = a 1,i−1 (i > 1) (7)a ij = a i+1,j−1 +a i,j−1 (j > 1); (8)• a ij = 0ifi−j ≠ 0,±2, a ii = λ i +λ i−1 (where λ 0 = λ n+1 = 0), a i+2,i = 1,a i,i+2 = λ 1 λ i+1 ;• a ij is the number of common divisors of i and j (2 ≤ i,j ≤ n);• a in = 0 if i+j is odd, a ij = ( 2kk)(i+j = 2k);• a ij = i+j−1j;• a ij = p j (x i ) where p j isapolynomial of degree j withleading coefficientc j and constant term 0;• a ij = j i−1 ;• a ij = x (i−1)(j−1) ;• a ij = p i+jq i+jwhere (p k ) and q k ) are arithmetic projections.VI.1.B Calculate the following determinants:6.2 General problems on determinantsWe continue with some more theoretical results involving determinants:29

VI.2.A Use the identity detAB = detAdetB for the matrices[ ] [ ]a −b c −dA = B =b a d cto prove the fact that if two integers areexpressible asthe sum of two squares(of integers), then the same is true of their product.VI.2.B Let B be an n × n matrix. What is the determinant f the linearmappingΦ : A ↦→ AB −BAon M n ?VI.2.C Show that if A is an (n+1)×n matrix with integer entries all ofwhose row sums are 0, show that det(AA t has the form nk 2 for some k ∈ N.VI.2.D LetA = [ A 1 A 2 ... A n]be an n×n matrix with columns A 1 ,...,A n . Show thatdet [ A 2 +···+A n A 1 +A 3 +···+A n ... A 1 +...A n−1]= (−1) n−1 (n−1)detA.]VI.2.E Show that if a matrix A is such that A 2 = I and B is the matrix[A ij ], then B 2 = I.VI.2.F Let Sx n 1 ,...,x k= ∑ α 1 +···+α k =n xα 11 ...x α k. Show thatS n x 1 ,...,x k= det[ a ]kdet [ a ].VI.2.G Show that if A is an n×n matrix, then|detA| ≤ ∏ ∑a jk .jShow that if for each i, a ii ≥ 1 2∑ nk=1 a ik, thenn∏detA ≥ a 11 (a ii −i=2kn∑|a ik A).k=130

VI.2.H Let (σ n ) be a sequence of real umbers and defineresp.Show that v(t)w(t) = 1.v(t) = 1−σ 1 t+w(t) = 1+σ 1 t+∞∑det [ a ] (−t) nn!n=2∞∑n=2det [ a ] t nn! .VI.2.I Show that if the elements of p + 2 columns of a matrix form anarithmetic progression of length p, then the determinant vanishes.VI.2.J Suppose that a 11 ,...,a 1n are integers. Show that one can find integersa ij (for 2 ≤ i ≤ n, 1 ≤ j ≤ n) so that det[a ij ] = 1 if and only if th egreatest common divisor of a 11 ,...,a 1n is 1.VI.2.K Let ξ 1 ,...,ξ n be real numbers so that∑ ∑ξ i = 0 ξi 2 = 0,..., ∑i i iξ n i = 0.Show that the ξ i vanish.VI.2.L Let A = [a ij ] be an n×n matrix. Show thatdet(I+A) = 1+n∑i=1⎡a ii + 1 [ ]2! det ai1 ,i 1i 1 ,i 2+···+ 1 a i2 ,i 1a i2 ,i 2 n! det ⎢⎣a i1 ,i 1.a in,i 1⎤... a i1 ,i n⎥. ⎦.... a in,i nVI.2.M Show that if the last element a nn of an n×n matrix is non-zero,then(a nn ) n−1 detA = detBwhere B is the (n−1)×(n−1) matrix withb ij = a ij a nn −a in a nj .(This provides an algorithm for calculating determinants by reducing thedimension step by step).31

VI.2.N Let Φ : L(V) → C be a linear mapping so that• Φ(f ◦g) = Φ(f)Φ(g);• Φ(λId) = λ nwhere V is a complex vector space of dimension n. Show that Φ(f) = detf.Show that the corresponding result for real vector spaces hold if n is odd butthat if n is even we require the additional condition that Φ(f) = −1 for somereflection in L(V) to conclude that Φ is the determinant.VI.2.O (Sylvester’s criterium)6.3 Determinants and smooth functionsIn this section, we gather some exercises on determinants involving derivativesof smooth functions. In particular, they can be used to give a particularlyelegant formulation of the chain rule for higher derivatives.IV.3.A Let x and y be continuously differentiable functions on the interval[a,b]. What well known result of analysis can one deduce by applying Rolle’stheorem to the functionVI.3.B Let⎡t ↦→ det⎣1 1 1x(t) x(a) x(b)y(t) y(a) y(b)⎤⎦x 1 = rc 1 c 2 ...c n−2 c n−1 (9)x 2 = rc 1 ...c n−2 s n−1 (10)vdots (11)x j = rc 1 ...c n−j s n−j+1 (12). (13)x n = rs 1 (14)where c i = cosθ i , s i = sinθ i (these are the equations of the transformationto polar coordinates in n dimensions). Calculate the Jacobi matrix∂(x 1 ,...,x n )∂(r,θ 1 ,...,θ n−1 )and show that its determinant is r n−1 c n−21 c n−32 ...c n−2 .32

VI.3.C Suppose that x is a smooth function on [a,b] and thatShow tht there is an s ∈]a,b[ so thata = t 0 < t 1 < ··· < t n = b.det [ a ] =??VI.3.D Show that if x 1 ,...,x n and y 1 ,...,y n are continuous functions on[a,b], then[∫ b]det x i (t)y j (t)dt = 1 ∫a n!∫...det[x i (t j )]dt 1 ...dt n .VI.3.E Let t ↦→ A(t) be a continuous function from R into the set M n ofn×n matrices and let X 1 ,...,X n be solutions of the equationShow that ifthenX ′ (t) = A(t)X(t).W(t) = det[X 1 (t)...X n (t)]∫ tW(t) = W(t 0 )exp trA(t)dt.t 0Show that if the function is continuously differentiable, thenddt detA(t) = tr(adjA(t))A′ (t)and thus if each A(t) is invertible, then this can be simplified to the formtr(A −1 (t)A ′ (t))detA(t).VI.3.F Let x 1 ,...,x n be linearly independent smooth functions on an interval,k ∈ N. Calculate⎡⎤x 1 (t) ... x n (t)lim h→0 h −k x 1 (t+h) ... x n (t+h)det⎢⎥⎣ . . ⎦ .x 1 (t+(n−1)h) ... x n (t+(n−1)h)33

VI.3.G Ifx 1 ,...,x n aresmoothfunctionson[0,1],theWronskianW(x 1 ,...,x n )is the function ⎡ ⎤x 1 (t) ... x n (t)x ′ 1t ↦→ det⎢(t) ... x′ n (t)⎥⎣ . . ⎦ .x (n)1 (t) ... x (n)n (t)Show that this either has no zeroooos or is identically zero. Show that if thex i are analytic, then the latter is the case f and only if the x i are linearlyindependent. Show that if x is a further function, thenW(xx 1 ,...,xx n ) = x n W(x 1 ,...,x n )and use this to evaluate the determinant⎡11 ...2! 12!det⎢⎣ .1n!1n!1 1...3! (n+1)!1 1...(n+1)! (2n−1)!.⎤⎥⎦ .VI.3.H Let x be a smooth function. Show that if⎡x ′ ⎤x 0 ... 0x ′′x ′ x ... 02 D n = det⎢⎥⎣ .. ⎦ ,x (n) x (n−1) x (n−2)... x ′ n! (n−1)! (n−2)!thenD n+1 = x ′ D n − 1n+1 x·D′ n .Show that if d r = 1 d r lnx, then(r−1)! dx⎡⎤d 1 −1 0 ... 0x (n) d 2 d 1 −2 ... 0= det⎢⎥⎣ . . ⎦ xn .d n d n−1 d n−2 ... s 1VI.3.I Show that if z is the composition of the smooth function x and y(i.e. z = x◦y), thenz (n) = det [ a ] .34

VI.3.J Let xbeafunctionwhich issmoothintheneighbourhoodofapointa and defineT n,a x(t) = x(a)+x ′ (a)(t−a)+···+ x(n) (a)(t−a) nn!(i.e. the Taylor approximation to x of degree n). Show that T n,a x(t) is equaltodet [ a ]and that7 Complex numbersf(x)−T n,a f(x) = det [ a ] .7.1 Complex numbers and geometryVII.1.A Show that if z 1 and z 2 are distinct complex numbers, then⎡ ⎤z ¯z 1det⎣z 1 z 1 1 ⎦ = 0z 2 z 2 1is the equation of the straight line through z −1 and z 2 .VII.1.B Let z 1 , z 2 and z 3 be complex numbers all with absolute value 1and putShow thats = z 1 +z 2 +z 3 (15)t = z 1 z 2 +z 2 z 3 +z 3 z 1 (16)p = z 1 z 2 z 3 . (17)• s 3 is the centroid of the triangle with vertices at z 1,z 2 and z 3 ;• orthocentre• circumcentreShow that the Euler line of the trianlge is{z : z = ¯s z} = {z : Is¯z = 0}sand that the nine-0nt circle has equationz = 1 4 (z − s 2 )−1 + ¯s 2 .35

VII.1.C Let z 0 , z 1 and z 2 be distinct complex numbers and defineR(z 0 ,z 1 ,z 2 ) = z 2 −z 0z 1 −z 0.Show that the numbers are collinear if and only if this is real. Similarly, wedefine CR(z 0 ,z 1 ;z 2 ,z 3 ) (the cross-ratio) of four distinct points to beR(y 0 ,y 1 ,y 2 )R(z 0 ,z 1 ,z 3 ) .Show that this is real if andonly if the points lie on a circle or a straight line.VII.1.D Show that if C is the circleaz¯b+bz +¯b¯z +c = 0then c is the power of 0 with respect to the circle. More generally, the poweraof w isw¯w+ b a w − ¯ba ¯w+ c a .Deduce that the locus of the set of points with constant power with respectto a given circle is itself a circle. If C and C 1 are two circles, the locus ofthe points which have the same power with respect to C and C 1 is a straightline which passes through the intersection of C and C 1 (if they intersect).VII.1.E Let C be the circleand define the mapping φ by(z −a)(z −a) = r 2φ(z) = r2¯z −āi.e. φ(z) is the inverse of z in C. Show that• C = {z ∈ C : φ(z) = z};• if z 1 , z 2 , z 3 , z 4 are points which are collinear with a, thenCR(z 1 ,z 2 ,z 3 ,z 4 ) = CR(φ(z 1 ),φ(z 2 ),φ(z 3 ),φ(z 4 ));• the mapping φ maps circles and straight line onto circles and straightlines;36

• a circle C 1 cuts C at right angles if and only if φ(C 1 ) = C 1 ;• if C 1 = φ(C 2 ) for two circles C 1 and C 2 , then a is a centre of similitudefor C 1 and C 2 (i.e.);• if C 1 and C 2 are two circles which are not concentric and have distinctradii, then there is a circle C as above so that C 2 = φ(C 2 ).VII.1.F Show that if z 1 ,z 2 , z 3 and z 4 are complex numbers, then(z 1 −z 4 )(z 2 −z 3 )+(z 2 −z 4 )(z 3 −z 1 )+(z 3 −z 4 )(z 1 −z 2 ) = 0.Deduce that if A, B, C and D are four points in the plane, then|AD||BC| ≤ |BD||CA|+|CD||AB.VII.1.G Let z 1 ,...,z n be points on the unit circle. Show that∏|z i z j −1| ≤ n ni≠jwith equality if and only if the z i are the vertices of a regular n-gon.7.2 Complex numbers and quaternionscomplex numbers also allow an elegant approach to the theory of quaternionsas the next examples show:VII.2.A Consider the set ˜Q of ordered pairs (z,w) of complex numberswith the natural addition and multiplication defied as follows:(z 0 ,w 0 )(z 1 ,w 1 ) = (z 0 z 1 −w 0 w 1 ,z 0 w 1 +z 1 w 0 ).Show that they satisfy all of the axioms of a field with the exception of thecommutativity of multiplication (such structures are called skew fields).VII.2.B ShowthatthereisanaturalisomorphismfromthesetQofquaternionsand the set ˜Q above whereby we map i onto (i,0), j onto (0,i) andkk onto ??VII.2.C A quaternion of the form (x,0) (x ∈ R) is said to be real. Showthat a quaternion is real if and only if it commutes with all quaternions.37

VII.2.D A double number is a number z = x+ey whereby e 2 = 1. Moreformally, the set of double nnumbers is R 2 provided with the multiplicationThen we defineRx = x Iz = y ¯z = x−ey z · ¯z = x 2 −y 2 .Show that7.3 PolynomialsVII.3.A Suppose that λ 1 , λ 2 and λ 3 are the roots of the cubicz 3 +pz 2 +qz +r = 0.Show thatSimilarly, show that if λ 1 ,...λ 4 are the roots ofthenz 4 +pz 3 +qz 2 +rz +s = 0VII.3.B Show that the roots of the equationare the vertices of a regular n-gon ifa 0 z n +a 1 z n−1 +···+ n = 0n k a k a k−10 =( nk)a k 1for ???? What are the special conditions for the case n = 1?VII.3.C Show that if p and q are polynomials with leading coefficients 1where f is of degree with distinct zero λ 1 ,...,λ n and g has degree n − 1,then ∑g(λi )f ′ (λ j ) = 1.38

VII.3.D Consider the polynomialwith zeroes λ 1 ,...,λ n andShwo that if z is a zero of p, thenp(z) = z n +a 1 z n−1 +···+a np ∗ (z) = z n +a ∗ 1 zn−1 +···+a ∗ n .|z| ≤ max{1, ∑ |a k |}.Show that if z ∗ is a root of p of multiplicity m and if ǫ > 0, then there isa δ > 0 so that if max|a n − a ∗ n| < δ, then p has m roots within ǫ of z ∗ .(Suppose that p k → p and that p k has less than m roots within ǫ of z ∗ . Usethe fact taht the roots are bounded and so have a convergent subsequence toget a contradiction.8 Eigenvalues, diagonalisation8.1 Eigenvalues of concrete matricesVIII.1.A Let A be a complex n×n matrix of the form⎡⎤a 1 0 ... 0 b 10 a 2 ... b 2 0⎢⎥⎣ . . ⎦0 ... 07 nb ni.e. where a ij = 0 if i+j ≠ n+1 and i−j ≠ n+1. Calculate the eigenvaluesof A. For which values of the a’s and b’s do its eigenvectors span C n ?VIII.1.B What are the eigenvalues of the matrix⎡⎤2a b 0 0 7... 0b 2a b 0 ... 0⎢⎥⎣ . . ⎦0 0 0 ... b 2aVIII.1.C Find the eigenvalues of the matrix A = [a ij ] where a ij = k ik jfora suitable sequence k 1 ,...,k n of non-zero numbers.39

8.2 Difference equationsThe method of diagonalising a matrix can be used to give an elegant treatmentofsomeproblemsinvolvingthedifferenceequations—forexample, questionsinvolving the Fibonacci numbers. We bring some examples of a similarnature.VIII.2.A Show that there is precisely one sequence (a n ) of non-negativenumbers which satisfies the recursion relationa 1 = 1 and a n+2 = a n −a n+1 .VIII.2.B Let (x n ) be a given sequence. For k > o, define a new sequenceby the relationsy 1 = x 1 (18)y n = kx n +x n−2 (n ≥ 2). (19)for which k do we have the following: the sequence (y n ) is convergent if anonly if (k n ) is?VIII.2.C Show that if f n is the n-th Fibonacci number, thenf 3 n+1 +f3 n +f4 n−1 = f 3n (20)f 3 n+2 −3f3 n +f3 n−2 =?f 3n. (21)VIII.2.D Outline a method for solving the difference equationApply it to the equationx n = a 1 x n−1 +a 2 x n−2 +···+a k x n−k .x n+1 = x n +x n−1 +···+x n−k+1 .VIII.2.E Define a sequence (t n ) by puttingt 1 = 2,t 2 = 3,t 2n =??Show that t 2n = βn−αn , t2 2n+1 = 2β n + 3αn where the α2 n and the β n are defineby the relationships [ ]2 1α n I +β n A =2 1where A =??.40

VIII.2.F Asequence (x n ) offunctions isdefined by therecurrence relation:x n (t) = tx n−1 (t)+x n−2 (t).Show thatx 2 n (t) ≤ (t2 +1) 2 (t 2 +2) n−3 .8.3 Location of eigenvaluesFor some applications it is important to have a priori estimates for theeigenvalues of a matrix (i.e. estimates involves the elements of the matrix).We bring two such estimates:VIII.3.A Show that if A is an n×n complex matrix, then its eigenvalueslie in the set: ⋃{λ ∈ C;|λ−a ii | ≤ ∑ |a ij |.ij≠iVIII.3.B Show that if A is a real n×n matrix, then its eigenvalue λ satisfythe inequality|Iλ| ≤ n(n−1)1/22max |a ij −a ji | 1/2 .1≤i,j≤n(This is a quantitative version of the fact that the eigenvalues of a symmetricmatrix are real).8.4 GeneralWe close this Chapter with some general properties of eigenvalues:VIII.4.A Show that if A is a complex N×n matrix and λ is an eigenvalue,then it is also an eigenvalue of adjA. Show thatχ adjA (λ) = (−1) n λndetA χ A( detAλ )and that if λ is not an eigenvalue of A, thentr(λI −A) −1 = χ′ A (λ)χ A (λ) .41

VIII.4.B Show that if A is an n×m matrix and B is an m ×n matrix,thenλ m χ AB (λ) = λ n χ BA (λ).What can you deduce about the relation between the eigenvalues of AB andthose of BA?VIII.4.C Show that if A has a block representation[ ]B C0 Dwhere B and C are square, then λ is an eigenvalue of A if and only if it isan eigenvalue of B or of D.VIII.4.D Let A be an n × n matrix, X a 1 × n and Y an n ×1 matrix.Show that the matrix [ ]A −AYB =−XA XAYis singular.Show that if detA = 0, then 0 is an eigenvalue of multiplicity at leasttwo of the above matrix.VIII.4.E Let P and Q be n × n matrices, where we assume that P isnon-singular and that A has distinct eigenvalues. Show that the polynomialsdet(λI−Q) and det(λP −PQ) have the same non-zeroes. Hence give a newproof of the fact thatdet(PQ) = detQ·detQ.Show that the latter equation can be extended to general P and Q by usinga continuity argument.VIII.4.F Suppose that A is ann×n matrix with rankr. Then aswe know,A can be expressed as a product BC whereB = [ B 1 0 ],C =[C10where B 1 is an n×r matrix and C 1 is r×n. Show that]χ A (λ) = λ n−r χ D (λ)where D = C 1 B 1 . Use this to calculate χ A where A = [b i c j ].42

VIII.4.G LetX = [ X 1 ... X n]be a non-singular n×n complex matrix (i.e. the X are the columns of X)and putY = [ X 2 X 3 ... X n 0 ]Show that Y X −1 has rank n−1 and that 0 is its only eigenvalue. Show thatif A is a matrix with rank n−1 and 0 as its only eigenvalue, then it has theabove form.VIII.4.H Let A andB be fixed n×n matrices. Show that there is a matrixX so thatX 2 −2AX +B = 0provided that the matrix [has 2n distinct eigenvalues.A IA 2 −B A]VIII.4.I Let A and B be fixed n×n matrices. Show that the operatorΦ : X ↦→ AX +XBis a linear operator on M n with eigenvalues {λ i +µ j } where {λ i } resp. {µ j }are the eigenvalues of A resp. B. Deduce a criterium for Φ to be invertible.VIII.4.J Show that if A and B are commuting n×n matrices and p is apolynomial in two variables, then the eigenvalues of p(A,B) are the numbersp(λ j ,µ j ) for some numbering λ 1 ,...,λ n resp., µ 1 ,...,µ n of the eigenvaluesof A and B.VIII.4.K Let A and B be complex n × n matrices so that AB = ωBAwhere ω is the primitive q-th root of unity. Show that if λ is an eigenvalueof A with eigenvector X so that BX ≠ 0, then ωλ is an eigenvalue of B.Show that the eigenvalues of A resp. B can be numbered as λ 1 ,...,λ nresp. µ 1 ,...,µ n in such a way that the eigenvalues of A+B resp. AB are{(λ q i +µq i )1/q : i = 1,...,n}resp.{ω q−12 λi µ i : i = 1,...,n}.43

9 The Jordan form9.1 The functional calculusOne of the most fundamental consequences of the existence of the Jordanform is the fact that every operator f has a unique representation f = d+nwhere d is diagonalisable, n is nilpotent and d and n commute. The if x is afunction which is such that ???? we definex(f) = ∑ kn∑ (f −λ k Id) rx (r) (λ k )E(λ k ) =r!r=0n∑∫x (r) (λ)Edλ.r=1σ(T)IX.A.1 Show that if the eigenvalues of f are distinct, thenx(f) = ∑ kx(λ k ) ∏ k≠jf −λ j Idλ k −λ j.IX.A.? A particularly important case is that where x is the exponentialfunction, when we writee f resp. e A forthecorresponding operatoror matrix.Show that if A has precisely one eigenvalue λ, thene A = e λ n∑k=01k! (A−λI)k .Show that if A has n distinct eigenvalues λ 1 ,...,λ n , thene A = ∑ e λ kL k (A)whereL k (t) = ∏ j≠kt−λ jλ k −λ j.Use this to calculate e A whereresp.A =A =[cosθ −sinθsinθ cosθ[cosθ sinθsinθ −cosθresp. e D where D is the differentiation operator on Pol(n).44]]

IX.A.? Verify the following identities:• e A+B = e A ·e B when A and B commute;• d dt etA = A·e tA ;• e A = lim n→∞ (I + A n )n ;• det(expA) = exp(trA).IX.A.4 Show that if all of the entries of A (with the possible exception ofthe diagonal ones) are positive, then all of the entries of e A are positive.IX.A.5 Show that if A is an n×n matrix such that A 2 = −I, thenUse this to solve the equatione tA = I cost+A ∈ t.y ′′ +y = 0.IX.A.6 Show that the general solution of the equationdXdt= AX +Bwhere B is a continuous mapping form R into M n,1 with initial conditionX(0) = x 0 is given by the equationX(t) =use this to solve the system∫ t0e (t−s)A B(s)ds+e tA X 0 .dxdt = y− 12 + e−3t (22)dy= −x+ 7y− 20z (23)dtdz= x− −5z +cost. (24)dt45

IX.A.7 If A is an n×n matrix, define functions a ij by the equationShow thate tA = [a ij (t)].χ A (D)a ij (t) = 0where D is the differentiation operator.(This reduces the calculation of e tA to the solution of the differentialequationsχ A (D)G(t) = 0with initial conditionsG(0) =), G ′ (0) = A,...,G (n−1 (0) = A n−1 .Use this to calculate e tA where⎡A = ⎣−2 −1 −16 3 2471 3⎤⎦.9.2 MiscellaneousWe conclude with a number of theoretical exercises where the Jordan formcan be used to advantage:IX.C.1 Let A be an n×n matrix with characteristic polynomial(−1) n χ A (λ) = λ n +a 1 λ n−1 +···+a n .Show thata k = − 1 k (a k−1S 1 +a k−2 S 2 +···+S k )where S k = tr(A k ). (This provides an algorithm for calculating the coefficientsof the characteristic polynomial without computing determinants).IX.C.2 If p is a polynomial of degree n, sayp(λ) = λ n +a 1 λ n−1 +···+a nwe put⎡A = ⎢⎣⎤0 1 0 ... 00 0 1 ... 0⎥.. ⎦ .−a n n −a n−1 −a n−2 ... −a 146

Aiscalled thecompanion matrix ofp. Showthat itscharacteristic polynomialis p and calculate a Jordan form for A. (consider first the special caseswhere the eigenvalues of A (i.e. the zeroes of p) are distinct, respectivelywhere they all coincide. Show how to use this result to solve the differentialequation p(D)x = 0.IX.C.3 LetAbeafixedn×nmatrix. Whatisthecharacteristicpolynomialof the mappingB ↦→ ABon M n ?IX.C.4 CharacterisethesetofallmatriceswhichcommutewiththeJordanmatrix J n (λ) resp. those matrices which commute with all matrices whichcommute with J n (λ). Use this to show that a matrix B is of the form p(A)where A is a fixed n×n matrix if and only if B commutes with every matrixwhich commutes with A.IX.C.5 Let λ be an eigenvalue of the operator f on V. DefineE λ = ⋃ kKer(f −λI) k .Wesaythatthenon-zerovector xinV isap-eigenvector off if(f−λI) p (x) =0but(f−λI) p−1 (x) ≠ 0. Showthatxisthenapeigenvector forf −1 providedthat the latter exists.IX.C.6 Let A be an n×n matrix with distinct10 Euclidean spacesX.1.A Show that if x,y,z are vector in a euclidean space, then‖x+y +z‖+‖x‖+‖y‖+‖z‖ ≥ ‖x+y‖+‖y +z‖+‖z +x‖.X.1.B Show that if (a 1 ,...,a n ) is a sequence of non-negative numbers,thenn∑ ( ai) 2n∑ a i a j ≤i i+j .i=1 i,j=147

X.1.C Show that if a 1 ≥ 0··· ≥ a n ≥ 0, then ∑ kj=1∑ kj=1 b j implies that∑ nj=1 a2 j j = 1n b 2 j .X.1.D Show that if (ξ i ) and (η i ) are vectors in R n , then( ∑ ξi 2 )(∑ ηi 2 )−(∑ ξ i η i ) 2 = ∑(ξ i η j −ξ j η i ) 2 .i≤i