Exercise 2.2. Find (with proof) all rational points on the curve y = 2 x.

Transcription

1 Exercise 2.2. Find (with proof) all rational points on the curve y = 2 x.

2 Exercise 2.4: Prove that for any integer n 0, e n is not a rational number.

3 Exercise 2.5: Prove that the only rational point on the curve y = e x is (0,1).

4 Exercise 2.6. Find (with proof) a point on the unit circle that is not a rational point.

5 Exercise 2.7. If (x, y) and (u, v) are rational points, show that the slope of the line between them is either rational or undefined.

6 Exercise 2.8. Show that: Adding two rational numbers always results in a rational number. Subtracting two rational numbers always results in a rational number. Multiplying two rational numbers always results in a rational number. Underwhat conditions does dividing two rational numbers result in a rational number?

7 Exercise 2.9. Express the statement in Exercise 2.7 in the form A implies B. What are the statements A and B? What is the converse of the statement? What is the contrapositive?

8 Exercise Consider a line through (0, -1) with slope 2/3. Show that this line intersects the unit circle in two points. Are they both rational? Consider a different line through (0, -1), this time with a rational slope of your own choosing. Show that the line intersects the unit circle in two points. Are they both rational? State and prove a conjecture suggested by these two exercises.

9 Exercise A straight line L is drawn through (0, -1) with slope p/q (p and q relatively prime integers). If p 0 then L intersects the unit circle in two points. Find both of them.

10 Proposition (r, s) is a rational point on the unit circle with r 0 iff r = 2pq/(p 2 +q 2 ) and s = (p 2 q 2 )/(p 2 +q 2 ) for some relatively prime integers p and q.

11 Proposition (a, b, c) is a nontrivial Pythagorean Triple if and only if (a/c, b/c) is a rational point on the unit circle.

12 Exercise Find 6 rational points on the unit circle using Proposition 2.14, and then use each to find a nontrivial Pythagorean Triple.

13 Proposition If p and q are relatively prime odd integers, then (pq, (p 2 q 2 )/2, (p 2 + q 2 )/2) is a primitive Pythagorean triple.

14 Proposition If p and q are relatively prime integers, one of which is even, then (2pq, p 2 q 2, p 2 + q 2 ) is a primitive Pythagorean triple.

15 Proposition Every primitive Pythagorean triple is either of the form (pq, (p 2 q 2 )/2, (p 2 + q 2 )/2) where p and q are relatively prime odd integers, or else it is of the form (2pq, p 2 q 2, p 2 + q 2 ) where p and q are relatively prime integers, one of which is even.

16 Exercise Explain how the preceding three theorems, when combined with the work of Harvey, Hymernie, and Rong, can be used to generate ALL possible Pythagorean triples.

17 Exercise For any open interval A = (s, t), let B = A. Show B is dense in A.

18 Exercise In the real plane, for any real x, let I x = {(x,y) 0 y 1}. Each I x is a vertical line segment connecting (x, 0) to (x, 1). Similarly, the set J = {(x,0) 0 x 1} is a horizontal line segment from (0, 0) to (1, 0). Define two sets as follows. A = J I 0 I 1 I 1/2 I 1/3 I 1/4 and B = J I 1 I 1/2 I 1/3 I 1/4 so B is a subset of A consisting of everything except the points in I 0. In the illustration at right, A is shown in red, and B is all of the points of A except the points on the y axis. The point of this example is that B is dense in A when we measure distances within the plane, but B is not dense in A if we measure distances within A. (Verify these claims!)

19 Proposition The rational points of the unit circle are a dense subset of the unit circle.

20 Exercise Find all integer solutions to the equation a 2 2b 2 = c 2. Include a proof that your answer includes all possible solutions. Note: this is an example of a Pell equation, a well known category of Diophantine equations.

21 Exercise Find all integer solutions to the equation a 2 + 5b 2 proof that your answer includes all possible solutions. = c 2. Again, include a

22 Exercise Find 5 more integer points on the curve with positive y coordinates. If you use a calculator or computing equipment, be sure to verify that your results are exact. (109,1138) might appear to be an integer point, but it is not exactly on C.

23 Exercise Show that a line through (2,5) with rational slope does not intersect C in other rational points.

24 Exercise Let L be the line through (-2,3) and (2,5). Show that this line intersects C in a third point, and that this point is rational.

25 Proposition For a cubic equation x 3 + ax 2 + bx + c = 0, the sum of the roots equals a.

26 Proposition Let (x 1, y 1 ) and (x 2, y 2 ) be distinct rational points on C. Let m be the slope of the line through these two points. Then there are rational points (x, ±y) on the curve with x = m 2 x 1 x 2.

27 Exercise Find three non-integer rational points on C.

28 Proposition 3.2 For any complex numbers z and w: i. For z 6= 0 6= w arg(wz) = arg(w)+arg(z) if that is in [0; 2ß), or arg(wz) = arg(w) + arg(z) 2ß otherwise. (That is, arg(wz) is equivalent to arg(w) + arg(z) mod 2ß.) ii. jzwj = jzjjwj. iii. jzj =0if and only if z =0. iv. <(z) =(z + z)=2. v. =(z) =(z z)=2i. vi. z 2 R iff z = z. vii. z = z. viii. z + w = z + w. ix. z w = z w. x. If p(t) is a polynomial with real coefficients, and if p(z) = w, then p(z) =w. xi. If p(t) is a polynomial with real coefficients, and if p(z) = 0, then p(z) =0. xii. 1=z =(1=jzj 2 )z for z 6= 0:

29 Exercise 3.4 Find all second, third, and fourth roots of unity. Express these in polar coordinates, and locate them on a diagram of the complex plane. Formulate and prove a conjecture about the n th roots of unity for any positive integer n:

30 Proposition 3.5 For any positive integer n, let U n be the set of all n th roots of unity. Then the following conditions hold: i. 1 2 U n ii. If z 2 U n then 1=z 2 U n too. iii. If z 2 U n and w 2 U n, then zw 2 U n too.

31 Exercise 3.7 Let n = 4: Show that both i and i are primitive roots of unity, while 1 and 1 are 4 th roots of unity that are not primitive. 4 th

32 Proposition 3.8 Let =2ß=n: Let z = e i. Then for 1» k» n 1; z k is a primitive n th root of unity if and only if k and n are relatively prime (meaning that they have no common divisors).

33 Exercise 3.9 Let w be any nonzero complex number. Find all solutions of the equation z n = w. [Hint: assume w = re i and express the solutions in polar form.]

34 Proposition 4.3 If is an eigenvalue of A, then the zero vector is an associated eigenvector.

35 Proposition 4.5 Let be an eigenvalue of A: Then E is closed under addition and scalar multiplication. That is, if x and y are elements of E and if s is a scalar, then x + y 2 E and sx 2 E.

36 Proposition 4.6 Let A be ann n matrix and let be ascalar. Then is an eigenvalue of A iff there exists a nonzero vector x such that ( I A)x =0:

37 Proposition 4.7 Let A be ann n matrix and let be ascalar. Then is an eigenvalue of A iff det( I A) =0:

38 Exercise 4.8 Find all the eigenvalues of the matrix A = For each eigenvalue find a parametric description of the corresponding eigenspace. 3 5 :

39 Proposition 4.11 If A is an n n triangular matrix with entries a ij, then the eigenvalues of A are the diagonal entries a kk for k =1; 2; ;n:

40 Proposition 4.12 Let A be ann n matrix. Let 1 ; 2 ; ; k be distinct eigenvalues of A: And let x 1 ; x 2 ; ; x k be nonzero eigenvectors associated with these eigenvalues, respectively. Then the set fx 1 ; x 2 ; ; x k g is linearly independent.

41 Proposition 4.13 Suppose A is an n n matrix, and that Ax j = j x j for j = 1; 2; ;m. Let P be the n m matrix with columns x 1 ; x 2 ; ; x m. Let D be the diagonal m m matrix with diagonal entries 1 ; 2 ; ; m : Then AP = PD:

42 Proposition 4.14 If A is an n n matrix with n distinct eigenvalues, there exists an invertible n n matrix P and an n n diagonal matrix D such that A = PDP 1. Moreover, the diagonal entries of D are eigenvalues and the corresponding columns of P are their respective associated eigenvectors.

43 Proposition 4.16 Every n n diagonal matrix is diagonalizable.

44 Proposition 4.17 If A = PDP 1 where P is an invertible n n matrix and D is an n n diagonal matrix, then the diagonal entries of D are eigenvalues and the corresponding columns of P are their respective associated eigenvectors.

45 Proposition 4.18 An n n matrix A is diagonalizable iff there exists a linearly independent set of n eigenvectors.

46 Exercise 4.19 The matrix is not diagonalizable. A = :

47 Exercise 4.20 The matrix is diagonalizable. A = :

48 Exercise 4.21 For any n 2 N and any scalar a, the n n matrix is not diagonalizable. A = a a a a a :

49 Exercise 4.22 For any n 2 N the n n matrix is diagonalizable. W n = :

50 Exercise 4.23 Give an example of an n n matrix (with n at least 5) that has repeated eigenvalues and is diagonalizable.

51 Proposition 4.25 Suppose A is a real matrix, and that is an eigenvalue with associated eigenvector v: Then is also an eigenvalue, and v is an associated eigenvector.

52 Proposition 4.26 If A is a real symmetric matrix, the eigenvalues are all real.

53 Proposition 4.27 Suppose A is a real symmetric n n matrix, and that 1 ; 2 ; ; k are distinct eigenvalues of A: And let x 1 ; x 2 ; ; x k be nonzero eigenvectors associated with these eigenvalues, respectively. Then for any 1» i<j» k; x i x j =0:

54 Proposition 4.29 Let A be a real n n symmetric matrix with distinct eigenvalues 1 ; 2 ; ; n. Then there exists an n n orthogonal matrix Q and a real n n diagonal matrix D such that A = QDQ T.

55 Exercise 5.1 The matrix A in (5.1) can be expressed as ai + bw + cv + du where I is the 4 4 identity matrix, and W;V; and U are specific numerical matrices. Find these matrices.

56 Exercise 5.2 For the W in the preceding exercise, and for any vector x =[x 1 x 2 x 3 x 4 ] T, show that W x =[x 2 x 3 x 4 x 1 ] T.

57 Exercise 5.3 Let A be a 4 n matrix. Describe the effect of multiplying A by W, with W on the left (that is, forming WA), in terms of the rows of A. Let B be anm 4 matrix. Describe the effect of multiplying B by W with W on the right (that is, forming BW), in terms of the columns of B. Taking A and B to be the identity matrix, describe W in terms of an identity matrix with permuted rows or columns.

58 Exercise 5.4 For the matrices in Exercise 5.1, show that V = W 2 and U = W 3.

59 Proposition 5.6 A 4 4 matrix A is a circulant matrix iff there exists a polynomial q(t) of degree 3 such that A = q(w ).

60 Exercise 5.9 Describe the effect of pre- or post-multiplying a matrix by W n. Then describe the powers of W n. Finally, describe the pattern of entries in an n n circulant matrix.

61 Exercise 5.10 Suppose D is an n n diagonal matrix with diagonal entries d k for k =1; 2; ;n; and q(t) isapolynomial. State and prove a proposition describing the matrix q(d).

62 Proposition 5.11 If A is an n n diagonalizable matrix, so that A = PDP 1 for some invertible matrix P and diagonal matrix D, and if q(t) is a polynomial, then q(a) =Pq(D)P 1

63 Proposition 5.12 If is an eigenvalue of A with x a corresponding eigenvector, then q( ) is an eigenvalue of q(a) and the same vector x is a corresponding eigenvector.

64 Exercise 5.13 Investigate whether the converse of the preceding proposition is true. That is, if B = q(a) and if B has an eigenvalue μ, must μ = q( ) for some eigenvalue of A? Put another way, could B have some eigenvalues in addition to the ones that we obtain by applying q to the eigenvalues of A? Could B have some other eigenvectors besides the ones we already know about as eigenvectors of A?

65 Proposition 5.14 Let A be an n n diagonalizable matrix, and let q(t) be a polynomial. If μ is an eigenvalue of q(a); then μ = q( ) for some eigenvalue of A.

66 Proposition 5.15 The eigenvalues of W n are the n th roots of unity.

67 Proposition 5.16 For any n 2 N, the matrix W n is diagonalizable.

68 Proposition 5.17 The eigenvalues of the circulant matrix q(w n ) are the complex numbers q(1);q(!);q(! 2 ); ;q(! n 1 ) where! = e 2ßi=n (or any other primitive n th root of unity).

69 Exercise 5.18 Find the eigenvalues of :

70 » ff fi Exercise 6.1 The general 2 2 circulant matrix is C = fi ff where ff and fi can be any scalars. Find the characteristic polynomial for C.

71 Exercise 6.2 Find ff and fi such that the characteristic polynomial of C is equal to f( ). Let this specific C be denoted C f.

72 Exercise 6.3 Use Propositions to find the eigenvalues of C f, and hence the roots of the original polynomial f(t).

73 Exercise 6.4 The preceding exercises show how roots of a specific quadratic can be found using circulant matrices. For a general (monic) quadratic, take f(t) =t 2 + bt + c, and repeat the same process as before, to derive an equivalent version of the quadratic formula.

74 Exercise 6.5 Find the characteristic polynomial for C.

75 Exercise 6.6 Find ff, fi, and fl such that the characteristic polynomial of C is equal to f( ). Let this specific C be denoted C f.

76 Exercise 6.7 Use Propositions to find the eigenvalues of C f, and hence the roots of the original polynomial f(t).

77 Proposition 6.8 The scalars r and s are the roots of t 2 + bt + c =0 if and only if r + s = b and rs = c.

78 Proposition 6.10 Let g(t) =t 3 +rt 2 +st+t, and let h(u) =g(u r=3). Then h(u) is a reduced cubic.

79 Proposition 6.11 The characteristic polynomial of the general 3 3 circulant C is a reduced cubic iff ff =0:

80 Exercise 6.12 The general 3 3 reduced circulant matrix is C 0 = fi fl fl 0 fi fi fl 0 where fi and fl can be any scalars. What is the characteristic polynomial for C 0? 3 5 :

81 Exercise 6.13 Let f(t) =t 3 +bt+c. Find fi and fl (expressed in terms of b and c) such that the characteristic polynomial of C 0 is identical to f( ).

82 Exercise 6.14 State and prove a theorem specifying the three roots of the cubic t 3 + bt + c in terms of the coefficients b and c.

83 Exercise 6.15 The general 4 4 reduced circulant matrix is C 0 = fi fl ffi ffi 0 fi fl fl ffi 0 fi fi fl ffi : where fi, fl and ffi can be any scalars. What is the characteristic polynomial for C 0?

84 Exercise 6.16 Let f(t) = t 4 + bt 2 + ct + d. Analyze the problem of choosing fi, fl and ffi so that the characteristic polynomial of C 0 is identical to f( ). In particular, set up a system of equations for fi, fl and ffi in terms of b; c; and d, and show that it leads to a cubic equation for fl 2 in terms of b; c; and d. If r is a root of this cubic, show how that leads to values of fi, fl and ffi. Finally, give equations for the roots of f in terms of b; c; d and r.

85 Proposition 6.17 Let f(t) =t 4 + bt 2 + ct + d, and let r be a root t 3 + b b 2 2 t d t c =0: Define fl = p r fi = s c + p c 2 4b 2 fl 2 16bfl 4 16fl 6 ffi = b +2fl2 : 4fi Then the roots of f are given by fl + fi + ffi; fl fi ffi; fl +(fi ffi)i; and fl (fi ffi)i: 8fl

86 Exercise 6.18 Apply the preceding result to find the roots of t 4 10t 2 60t +144=0; verifying that your answers are correct.

87 Proposition 7.1 Suppose that a polynomial p(t) = t n + a n 1 t n a 1 t + a 0 has roots r 1 ;r 2 ; ;r n. Then a n 1 = (r 1 + r r n ) and a 0 =( 1) n (r 1 r 2 r n ).

88 Exercise 7.2 Suppose p(t) =(t u)(t v)(t w). Expand p(t) to the standard descending form. How does the coefficient of t depend on u; v; and w?

89 Proposition 7.3 The scalars r;s; and u are the roots of t 3 + bt 2 + ct + d =0if and only if r + s + u = b rs + ru + su = c rsu = d:

90 Exercise 7.4 Suppose p(t) =(t u)(t v)(t w)(t x). Expand p(t) to the standard descending form. How do the coefficients of t 2 and t depend on u; v; w; and x?

91 Exercise 7.5 Suppose p(t) = (t r 1 )(t r 2 ) (t r n ). If p(t) is expanded tothe standard descending form t n + a n 1 t n a 1 t + a 0 formulate a conjecture expressing each a j in terms of r 1 ;r 2 ; ;r n :

92 Exercise 7.7 Suppose p(t) =(t r 1 )(t r 2 ) (t r n ). Suppose p(t) is expanded to the standard descending form t n + a n 1 t n a 1 t + a 0. Give an equation for each a k ;k = 0; 1; ;n 1; in terms of the elementary symmetric functions ff j (r 1 ;r 2 ; ;r n ).

93 Exercise 7.8 The sum of the squares of the roots of t n + a n 1 t n a 1 t + a 0 is given by a 2 n 1 2a n 2 :

94 Exercise 7.9 For the variables x 1 ;x 2 ; ;x n ; express nx j=1 x 3 j in terms of the elementary symmetric functions. Then use this result to find a formula for the sum of the cubes of the roots of a polynomial in terms of the coefficients of the polynomial.

95 Definition 7.10 For each k 2 N, let s k = r k 1 + r k rn k. Also, let S(t) = 1X k=0 s k t k.

96 Proposition 7.11 Let m be the maximum of jr 1 j; jr 2 j; ; jr n j: Then S(t) converges absolutely for 1=m < t < 1=m, and is given by S(t) = 1 1 r 1 t r 2 t r n t :

97 Proposition 7.12 Let h(t) =p 0 (t)=p(t). Then and h(t) = 1 t r t r t r n 1 1 t h = t 1 1 r 1 t r 2 t r n t :

98 Proposition 7.13 If g(t) is a polynomial of degree d, then rev g(t) = t d g(1=t).

99 Proposition 7.14 For any t 62 f0; 1=r 1 ; 1=r 2 ; ; 1=r n g, rev p 0 (t) rev p(t) = p0 (1=t) tp(1=t) = 1 1 t h : t

100 Proposition 7.15 Let m be the maximum of jr 1 j; jr 2 j; ; jr n j: Then for 1=m<t<1=m, S(t) = rev p0 (t) rev p(t) :

101 Proposition 7.16 If a 0 roots of p is given by 6= 0; then the sum of the reciprocals of the 1 r r r n = a 1 a 0 :

102 Proposition 7.17 The average of the roots of p is given by a n 1 =n.

103 Proposition 7.18 If p(t) is any polynomials, the average of the roots of p is equal to the average of the roots of its derivative, p 0.

104 Exercise 8.2 The permutation matrix W n is a companion matrix. Find the polynomial p such that W n = C p :

105 Exercise 8.3 Make up examples of companion matrices for polynomials of degree 2, 3, 4. For each one, find the characteristic polynomial. Then state and prove a theorem about the characteristic polynomial of a companion matrix.

106 Proposition 8.4 Suppose an n n matrix X is partitioned as " # A B C D where A and D are square and invertible, and C is a zero-matrix. Then X is invertible, and the inverse is given by " A 1 A 1 BD 1 0 D 1 # :

107 Exercise 8.5 Use the preceding proposition to find the inverse of :

108 Exercise 8.6 Suppose we have a partitioned matrix in the form " # A B X = ; R and we want to multiply by matrix Y. Show that Y can be partitioned vertically into blocks Y 1 and Y 2 such that " # AY1 + BY 2 XY = : RY

109 Proposition 8.7 For a scalar c, let v =[1 c c 2 p(t) =t n + a n 1 t n a 1 t + a 0. Then C p v = c c 2. c n 1 c n p(c) : c n 1 ] T, and let

110 Proposition 8.8 Let v(t) = [1 t t 2 t n 1 ] T, and let p(t) = t n + a n 1 t n a 1 t + a 0. Then C p d dt v(t) = d dt t t 2. t n 1 t n p(t) :

111 Exercise 8.9 Let p(t) =(t 2)(t 3)(t 4), and consider the companion matrix C p : Find the eigenvalues and eigenvectors for C p : Formulate and prove a conjecture about the eigenvalues and eigenvectors of companion matrices.

112 Exercise 8.10 If p(t) has distinct roots, show C p is diagonalizable and find the diagonalization.

113 Proposition 8.13 The determinant of a vandermonde matrix is given by Y det V (r 1 ;r 2 ; ;r n )= (r k r j ): 1»j<k»n

114 Proposition 8.14 A vandermonde matrix V (r 1 ;r 2 ; ;r n ) is invertible iff the scalars r 1 ;r 2 ; ;r n are distinct.

115 Exercise 8.15 If r 1 ;r 2 ; ;r n are distinct, the matrix equation V (r 1 ;r 2 ; ;r n ) x 1 x 2. x n 1 x n = has the unique solution x 1 x 2. x n = p 0 (r 1 ) 1 p 0 (r 2 ). 1 p 0 (r n) where p(t) =(t r 1 )(t r 2 ) (t r n ).

116 Proposition 8.16 Let (x 1 ;y 1 ); (x 2 ;y 2 ); ; (x n ;y n ) be n points in C 2, with distinct x values. Then i. The polynomial a n 1 x n 1 + a n 2 x n a 0 interpolates the points (x k ;y k ) iff V (x 1 ;x 2 ; ;x n ) T a 0 a 1. a n = ii. There exists a unique such polynomial y 1 y 2. y n :

117 Proposition 8.17 Let f(t) be a monic polynomial of degree n: There exists a circulant matrix C whose characteristic polynomial, p C ( ) equals f( ).

118 Exercise 9.2 The Fibonacci numbers are a solution to a second order homogeneous difference equation. Find a different solution to the same difference equation.

119 Exercise 9.4 What is the characteristic polynomial for the Fibonacci difference equation?

120 Proposition 9.5 Let n 2 N ; let a 0 ;a 1 ; ;a n 1 bescalars, and let f(k) be a function defined for k =0; 1; 2; : Finally, let s ffi 0 ; s ffi 1 ; ; s ffi n 1 be given scalars. Then there exists a unique sequence fs k g that satisfies (9.1) and such that s k = ffi s k for 0» k» n 1.

121 Exercise 9.7 Find the unique solution to the IVP s k+1 = as k ; s 0 = ffi s:

122 Proposition 9.9 The VIVP s k+1 = As k ; s 0 = s ffi has a unique solution, given by s k = A k s: ffi

123 Exercise 9.10 Let fs k g be a scalar sequence. Define a corresponding vector sequence fs k g by» sk s k = : s k+1 Show that the scalar sequence fs k g satisfies the IVP s k+2 = s k+1 + s k ; s 0 =0; s 1 =1 iff the vector sequence fs k g satisfies the VIVP»» s k+1 = s 1 1 k ; s 0 = 1 : Then show that F k ; the kth Fibonacci number, is given by» k» F k =[1 0] : 1 1 1

124 Proposition 9.11 For any k 2 N ; A k F =» Fk 1 F k F k F k+1 :

125 !!! ). '/

126 Proposition 9.13 The sum of the squares of two consecutive Fibonacci numbers is always another Fibonacci number. In fact, for any k 2 N ; F 2 k + F 2 k+1 = F 2k+1.

127 Exercise 9.14 Find an invertible matrix P and a diagonal matrix D such that A F = PDP 1. Then substitute that in the equation» 0 F k =[1 0]A k F 1 to derive a formula for F k :

128 Exercise 9.15 Let fs k g be a scalar sequence. Define a corresponding vector sequence fs k g by s k = s k s k+1. s k+n : a. Find an n n matrix A such that the scalar sequence fs k g satisfies the difference equation s k+n = a n 1 s k+n 1 + a n 2 s k+n a 0 s k iff the vector sequence fs k g satisfies the vector difference equation s k+1 = As k b. Let initial values s ffi k be defined for k =0; 1; ;n 1; and let the vector s ffi = [ s ffi ffi ffi 0 s1 s n 1 ] T. Show that a solution fs k g of the difference equation in part a: also satisfies the initial conditions iff the vector solution s k satisfies the initial condition s 0 = s. ffi c. The preceding parts together define an IVP. Show that the solution to the IVP is s k =[ ]A k ffi s: (9.4)

129 Proposition 9.16 Let fs k g be a solution to the difference equation s k+n = a n 1 s k+n 1 + a n 2 s k+n a 0 s k ; and suppose the characteristic polynomial has distinct roots r 1 ;r 2 ; ;r n. Then s k = c 1 r k 1 + c 2 r k c n r k n for some scalars c 1 ;c 2 ; ;c n.

130 Proposition 9.18 Suppose fs k g is a generalized Fibonacci number sequence with difference equation s k+n = a n 1 s k+n 1 + a n 2 s k+n a 0 s k ; and suppose that the characteristic polynomial has distinct roots r 1 ;r 2 ; ;r n. Then s k = rk 1 p 0 (r 1 ) + rk 2 p 0 (r 2 ) + + rk n p 0 (r n ) :

131 Exercise 9.19 Find a formula for the generalized Fibonacci number sequence with the following difference equations a. s k+3 = s k+2 s k+1 + s k b. s k+3 =2s k+2 + s k+1 2s k.

132 Exercise 10.2 Find all the eigenvalues and eigenvectors for an n n Jordan block matrix J(c).

133 Exercise 10.4 For any scalar c, J(c) =ci + N:

134 Proposition 10.5 Let A and B be n n matrices, such that AB = BA. Then, for k 2 N ; kx k (A + B) k = A j B k j : j j=0

135 Exercise 10.6 For any k 2 N (J(c)) k = kx j=0 k j c k j N j :

136 Exercise 10.7 Compute the powers of n nnmatrices with n =3; 4; and 5. Compute enough powers of each to detect a pattern in the results. Then formulate and prove a proposition giving the value of N k for any k:

137 Exercise 10.8 Use the preceding results to determine the pattern of entries in (J(c)) k :

138 Exercise 10.9 Use the prior results and block matrix multiplication to find J k for each of the example J matrices in (10.1).

139 Proposition If f(t) =(t r) n, then f hmi (t) = 8 < : n! (t r)n m if (n m)! m<n n! if m = n 0 if m>n:

140 Proposition Let f(t) and g(t) be functions whose derivatives exist up to f hmi (t) and g hmi (t). Then (f(t)g(t)) hmi exists and is given by mx m (f(t)g(t)) hmi = f hji (t)g hm ji (t): j j=0

141 Proposition A polynomial p has a root r of multiplicity m iff p hji (r) =0for j =0; 1; 2; ;m 1; and p hmi (r) 6= 0.

142 Proposition Let q(t) = a 0 + a 1 t + + a n 1 t n 1, and define the vector c = [a 0 a 1 a n 1 ] T. Then, q(t) = c T v(t) and for any m 2 N ; q hmi (t) =c T v hmi (t).

143 Proposition Let p(t) =t n + a n 1 t n a 1 t + a 0. Then d C p v hmi (t) = dt m t t 2. t n 1 t n p(t) 3 d 7 5 = dt m (tv(t)) p hmi (t) :

144 Exercise Using Proposition 10.12, show that m d (tv(t)) = tv hmi (t)+mv hm 1i (t): dt Then, if r is a root of p of multiplicity m +1 or greater, show that C p v hmi (r) =rv hmi (r)+mv hm 1i (r)

145 Proposition Let p be a monic polynomial of degree n. Suppose that r isaroot of p of multiplicity n 1. Let P 1 be the n n 1 matrix with columns (1=j!)v hji (r), for j =0; 1; 2; ;n 1 1. Let J 1 be the n 1 n 1 Jordan block matrix J(r). Then C p P 1 = P 1 J 1 :

146 Proposition Let p(t) =(t r 1 ) n 1 (t r 2 ) n2 (t r s ) ns, where the roots r 1 ;r 2 ; ;r s are distinct, and the exponents n 1 ;n 2 ; ;n s are in N. Let n = n n s. For each root r i let J i be the n i n i Jordan block matrix J(r i ), and let P i bethen n i matrix with columns (1=j!)v hji (r i ), for j =0; 1; 2; ;n i 1. Let P be the partition matrix [P 1 jp 2 j jp s ], and let J be the block diagonal matrix whose ith diagonal block is J i : Then C p P = P J.

147 Proposition The matrix P in the preceding proposition is invertible, and C p = PJP 1 where J is a JCF matrix.

148 Exercise Let p(t) = (t 2) 3 (t 1) 2 : Find C p, P; and J, and verify the equation C p = PJP 1. Then find an equation for C k p :

149 Exercise At the end of Chapter 9 you used matrix methods to find generalized Fibonacci number formulas for a few difference equations. But those methods would not have worked for s k+3 = s k+2 + s k+1 s k. (Why?) This difference equation can now be analyzed using the results from this chapter. Find a formula for s k :

150 Proposition 11.2 Let p be a monorootic polynomial with root r. Then the companion matrix is given in partition form by " # 0 I C p = ; v T where 0 is the n 1-entry zero vector, I is an (n 1) (n 1) identity matrix, and v T is the row vector h n 0 ( r) n n 1 ( r) n 1 n 2 ( r) n 2 n n ( r) 1 i :

151 Exercise 11.3 In the case of a monorootic polynomial, the JCF of the companion matrix can be found using Proposition Find the matrices J and P of that proposition when p(t) =(t r) n for n =3; 4; 5: What pattern do you see? What is the pattern when r =1?

152 Proposition 11.6 For any polynomial q and scalars r and s the following hold: i. q 0 = q ii. (q r ) s = q r+s iii. (q r ) ( r) = q.

153 Proposition 11.8 Let q be a polynomial of degree m. Then for the (m +1) (m +1) P (r). [q r ]=P (r) T [q]

154 Proposition 11.9 For any n 2 N, the n n matrices P (r) satisfy the following: a. P (r)p (s)=p (r + s) b. P (0) = I c. P (r)p (s)=p (s)p (r) d. P (r) is invertible with P (r) 1 = P ( r).

155 Exercise Compare n n matrices P (r) and P ( r) for n = 3; 4; 5: Observe the pattern of the entries in P ( r), and verify in each case that P ( r) =P (r) 1.

156 Exercise 12.1 Formulate a conjecture about the sum k 3 : Prove or disprove it.

157 Exercise 12.3 Give the matrix equations for S hri k when r =1; 2 and 3. Expand out the first two equations to verify that each gives the correct formula for S hri k a formula for S h3i k.. Then expand the matrix equation for r =3to derive

158 Exercise 12.4 Show that S h2i k satisfies the difference equation s k+1 = s k + k 2 +2k +1: (12.1)

159 Proposition 12.6 The operator L is linear: for any sequences fs k g and ft k g and any scalars a and b, L (afs k g + bft k g)=alfs k g + blft k g:

160 Exercise 12.7 Show that a sequence fs k g satisfies (12.1) iff it satisfies (L 1)fs k g = k 2 +2k +1:

161 Exercise 12.8 For a fixed n 2 N and a function f(k), show that a sequence fs k g satisfies the difference equation s k+n = a n 1 s k+n 1 + a n 2 s k+n a 0 s k + f(k) iff it satisfies where p(l)fs k g = ff(k)g p(l) =L n a n 1 L n 1 a n 2 L n 2 a 0 is the characteristic polynomial of the difference equation.

162 Exercise Let f(k) = k 2 +2k +1: Show that q(l) = (L 1) 3 annihilates ff(k)g.

163 Exercise Show that S h2i k is the unique solution of the IVP (L 1) 4 fs k g =0; s 0 =0;s 1 =1;s 2 =5;s 3 =14: Then use the methods of Chapter 9 and Chapter 11 to derive a formula for S h2i k.

164 Proposition Let f(k) be a polynomial of degree m 0. Then ff(k)g is annihilated by (L 1) m+1 :

165 Proposition For any r 2 N, S hri k difference equation IVP is the unique solution to the (L 1) r+2 s k =0; s k =0 r +1 r + + k r for k =0; 1; ;r+1:

166 Proposition For any r 2 N ; S hri k S hri k =[1 0 0 ::: 0](C p ) k where p(t) =(t 1) r is given by r 1+2 r +3 r. 1+2 r +3 r + +(r +1) r 3 7 5

167 Proposition For any r 2 N ; S hri k h S hri k = 1 k 1 k 2 ::: i 6 k 6 r +1 P ( 1) is given by r 1+2 r +3 r. 1+2 r +3 r + +(r +1) r where P ( 1) is given by Definition 11.4.

168 Exercise The column vector above, [ ] T, can be expressed in the form Hw where H is a square matrix and w is the vector [ ] T. Find H.

169 Proposition For any r 2 N ; S hri k S hri k = h k 1 k 2 ::: k i r +1 is given by P ( 1) r 3 r. (r +1) r :

170 Proposition Let g(t) be a polynomial of degree r. Then kx j=1 g(j) = h k k 1 2 ::: k i r +1 P ( 1) g(1) g(2) g(3). g(r +1) :