Solutions hapter ilin Linear Iterative Systems Suppose u () = Find u (), u (), and u () when (a) u (k+) = u (k), (b) u (k+) = 9u (k), (c) u (k+) = i u (k), (d) u (k+) = ( i )u (k) Is the system (i) stable? (ii) asymptotically stable? (iii) unstable? (a) u () =, u () = and u () = 8576; unstable (b) u () = 9, u () = 8678 and u () = 577; asymptotically stable (c) u () = i, u () = and u () = ; stable (d) u () = i, u () = 7 + 6 i and u () = 96587 + 7698 i ; unstable bank offers 5% interest compounded yearly Suppose you deposit $ (a) Set up a linear iterative equation to represent your bank balance (b) How much money do you have after years? (c) What if the interest is compounded monthly? (a) u (k+) = 5 u (k), u () =, where u (k) represents the balance after k years u () = 5 = 769 dollars (b) u (k+) = ( + 5/) u (k) = 78 u (k), u () =, where u (k) represents the balance after k months u () = ( + 5/) = 8 dollars Show that the yearly balances of an account whose interest is compounded monthly satisfy a linear iterative system How is the effective yearly interest rate determined from the original annual interest rate? If r is the yearly interest rate, then u (k+) = ( + r/) u (k), where u (k) represents the balance after k months Thus, the balance after m years is v (m) = u ( m), and satisfies v (m+) = ( + r/) v (m) Then ( + r/) = + y where y is the effective annual interest rate Show that, as the time interval of compounding goes to zero, the bank balance after k years approaches an exponential function e r k a, where r is the yearly interest rate and a the initial balance The balance coming from compounding n times per year is + n«r n k a e r k a by a standard calculus limit, [] 5 Let u(t) denote the solution to the linear ordinary differential equation u = αu, u() = a Let h > be fixed Show that the sample values u (k) = u(k h) satisfy a linear iterative system What is the coefficient λ? ompare the stability properties of the differential equation and the corresponding iterative system Since u(t) = ae α t we have u (k+) = u (k + )h = ae α (k+) h = e α h ae α k h = iter 9/9/ 58 c Peter J Olver
e α h u (k), and so λ = e α h The stability properties are the same: α < for asymptotic stability; α for stability, α > for an unstable system 6 For which values of λ does the scalar iterative system () have a periodic solution, meaning that u (k+m) = u (k) for some m? The solution u (k) = λ k u () is periodic of period m if and only if λ m =, and hence λ is an m th root of unity Thus, λ = e i k π/m for k =,,, m If k and m have a common factor, then the solution is of smaller period, and so the solutions of period exactly m are when k is relatively prime to m and λ is a primitive m th root of unity, as defined in Exercise 577 7 Investigate the solutions of the linear iterative equation when λ is a complex number with λ =, and look for patterns You can take the initial condition a = for simplicity and just plot the phase (argument) θ (k) of the solution u (k) = e i θ(k) If θ is a rational multiple of π, the solution is periodic, as in Exercise 6 When θ/π is irrational, the iterates eventually fill up (ie, are dense in) the unit circle More? 8 onsider the iterative systems u (k+) = λ u (k) and v (k+) = µ v (k), where λ > µ Prove that, for any nonzero initial data u () = a, v () = b, the solution to the first is eventually larger (in modulus) than that of the second: u (k) > v (k), for k u (k) = λ k a > v (k) = µ k b provided k > inequality relies on the fact that log λ > log µ log b log a, where the log λ log µ 9 Let λ, c be fixed Solve the affine (or inhomogeneous linear) iterative equation u (k+) = λ u (k) + c, u () = a (5) Discuss the possible behaviors of the solutions Hint: Write the solution in the form u (k) = u + v (k), where u is the equilibrium solution The equilibrium solution is u = c/( λ) Then v (k) = u (k) u satisfies the homogeneous system v (k+) = λ v (k), and so v (k) = λ k v () = λ k (a u ) Thus, the solution to (5) is u (k) = λ k (a u ) + u If λ <, then the equilibrium is asymptotically stable, with u (k) u as k ; if λ =, it is stable, and solutions that start near u stay nearby; if λ >, it is unstable, and all non-equilibrium solutions become unbounded: u (k) bank offers 5% interest compounded yearly Suppose you deposit $ in the account each year Set up an affine iterative equation (5) to represent your bank balance How much money do you have after years? fter you retire in 5 years? fter years? Let u (k) represent the balance after k years Then u (k+) = 5 u (k) +, with u () = The equilibrium solution is u = /5 =, and so after k years the balance is u (k) = (5 k ) Then u () = $, 595, u (5) = $5, 76, u () = $, 9, 979 Redo Exercise in the case when the interest is compounded monthly and you deposit $ each month If u (k) represent the balance after k months, then u (k+) = ( + 5/) u (k) +, u () = The balance after k months is u (k) = (667 k ) Then u () = $, 558, u (6) = $6, 6865, u () = $5, 77, 7 Each spring the deer in Minnesota produce offspring at a rate of roughly times iter 9/9/ 59 c Peter J Olver
the total population, while approximately 5% of the population dies as a result of predators and natural causes In the fall hunters are allowed to shoot, 6 deer This winter the Department of Natural Resources (DNR) estimates that there are, deer Set up an affine iterative equation (5) to represent the deer population each subsequent year Solve the system and find the population in the next 5 years How many deer in the long term will there be? Using this information, formulate a reasonable policy of how many deer hunting licenses the DNR should allow each fall, assuming one kill per license u (k+) = 5 u (k) 6, u () = The equilibrium is u = 6/5 =, Since λ = 5 >, the equilibrium is unstable; if the initial number of deer is less than the equilibrium, the population will decrease to zero, while if it is greater, then the population will increase without limit Two possible options: ban hunting for years until the deer reaches equilibrium of, and then permit at the current rate again Or to keep the population at, allow hunting of only, deer per year In both cases, the instability of the equilibrium makes it unlikely that the population will maintain a stable number, so constant monitoring of the deer population is required (More realistic models incorporate nonlinear terms, and are less prone to such instabilities) Find the explicit formula for the solution to the following linear iterative systems: (a) u (k+) = u (k) v (k), v (k+) = u (k) + v (k), u () =, v () = (b) u (k+) = u (k) v(k), v (k+) = u(k) 6 v(k), u () =, v () = (c) u (k+) = u (k) v (k), v (k+) = u (k) + 5v (k), u () =, v () = (d) u (k+) = u(k) + v (k), v (k+) = v (k) w (k), w (k+) = w(k), u () =, v () =, w () = (e) u (k+) = u (k) + v (k) w (k), v (k+) = 6u (k) + 7v (k) w (k), w (k+) = 6u (k) + 6v (k) w (k), u () =, v () =, w () = (a) u (k) = k + ( ) k, v (k) = k + ( ) k (b) u (k) = k + 8 k, v(k) = 5 k + 8 k (c) u (k) = ( 5 + )( 5) k + ( 5 )( + 5) k, v (k) = ( 5) k ( + 5) k 5 5 (d) u (k) = 8 + 7 k 8 k, v(k) = + k, w(k) = k (e) u (k) = k, v (k) = + ( ) k k+, w (k) = ( ) k k Find the explicit formula for the general solution to the linear iterative systems with the following coefficient matrices: 5 7 6 6 (a), (b), (c) 6, (d) 6 5 (a) u (k) = c ( ) k + c ( + ) k ; iter 9/9/ 5 c Peter J Olver
(b) u (k) = c + i k 5 i + c i k 5+ i = a 5 cos k π + sin k π cos k π + a 5 sin k π cos k π ; sin k π (c) u (k) = c + c ( ) k + c ( ) k ; (d) u (k) = c k + c k + c 6 k 5 The k th Lucas number is defined as L (k) = + 5 + 5 (a) Explain why the Lucas numbers satisfy the Fibonacci iterative equation L (k+) = L (k+) +L (k) (b) Write down the first 7 Lucas numbers (c) Prove that every Lucas number is a positive integer (a) It suffices to note that the Lucas numbers are the general Fibonacci numbers (6) when a = L () =, b = L () = (b),,,, 7,, 8 (c) ecause the first two are integers and so, by induction, L (k+) = L (k+) + L (k) is an integer whenever L (k+), L (k) are integers 6 Prove that all the Fibonacci integers u (k), k, can be found by just computing the first term in the inet formula (7) and then rounding off to the nearest integer The second summand satisfies 5 5 k k k < 8 6 k < 5 for all k QED 7 What happens to the Fibonacci integers u (k) if we go backward in time, ie, for k <? How is u ( k) related to u (k)? u ( k) = ( ) k+ u (k) Indeed, since u ( k) = 6 + 5 5 = 6 ( ) k 5 k 5 = ( )k+ 6 + 5 5 5 k k + 5 ( ) k 5 = + 5 k 7 5 = 5 6 + 5 + 5 k 7 5 k 7 5 = ( ) k+ u (k), k 5 8 Use formula () to compute the k th power of the following matrices: 5 (a), (b), (c), (d), (e) k 7 5 iter 9/9/ 5 c Peter J Olver
(a) (b) (c) (d) (e) 5 k = 6 k k = k = k 5 + 5 i = k = i 5 5 5 5 k k, ( + i )k ( i ) k + 5 + 5 5 k ( ) k, i i «k + 5, 6 6, 5 5 «k 5+ 5 5 5 5+ 5 5+ 5 5 5 5 5 9 Use your answer from Exercise 8 to solve the following iterative systems: (a) u (k+) = 5u (k) + v (k), v (k+) = u (k) + v (k), u () =, v () =, (b) u (k+) = u (k) + v (k), v (k+) = u (k) + v (k), u () =, v () =, (c) u (k+) = u (k) + v (k), v (k+) = u (k) + v (k), u () =, v () =, (d) u (k+) = u (k) + v (k) + w (k), v (k+) = u (k) + v (k) + w (k), w (k+) = u (k) + v (k) + w (k), u () =, v () =, w () =, (e) u (k+) = v (k), v (k+) = w (k), w (k+) = u (k) + w (k), u () =, v () =, w () = (a) (c) (e) u (k) v (k) = i i u (k) v (k) = w (k) u (k) v (k) = 5 6 k 5 + 5 i ( + i )k ( i ) k + 5 5 5, (b), (d) 5+ 5 5+ 5 u (k) v (k) = u (k) v (k) = w (k) «k + 5 «k + 5 k k+, (a) Given initial data u () = (,, ) T, explain why the resulting solution u (k) to the system in Example 7 has all integer entries (b) Find the coefficients c, c, c in the explicit solution formula (8) (c) heck the first few iterates to convince yourself that the solution formula does, in spite of appearances, always give an integer value (a) Since the coefficient matrix T has all integer entries, its product T u with any vector with integer entries also has integer entries; (b) c =, c =, c = ; 6 76 6 (c) u () =, u () =, u () =, u () = 76, u (5) = 5 6 88 k, iter 9/9/ 5 c Peter J Olver
(a) Show how to convert the higher order linear iterative equation u (k+j) = c u (k+j ) + c u (k+j ) + + c j u (k) into a first order system u (k) = T u (k) Hint: See Example 6 (b) Write down initial conditions that guarantee a unique solution u (k) for all k The vectors u (k) = u (k), u (k+),, u (k+j ) T R j satisfy u (k+) = T u (k), where T = The initial conditions are u () = a = c j c j c j c j c T a, a,, a j, and so u () = a, u () = a,, u (j ) = a j pply the method of Exercise to solve the following iterative equations: (a) u (k+) = u (k+) + u (k), u () =, u () = (b) u (k+) = u (k+) + u (k), u () =, u () = (c) u (k+) = u (k+) + u (k), u () =, u () = (d) u (k+) = u (k+) u (k), u () =, u () = (e) u (k+) = u (k+) + u (k+) u (k), u () =, u () =, u () = (f ) u (k+) = u (k+) + u (k+) u (k), u () =, u () =, u () = (a) u (k) = ( )k, (b) u (k) = k + k, (c) u (k) = (5 5)( + 5) k + (5 + 5)( 5) k ; (d) u (k) = i ( + i ) k + + i ( i ) k = k/ cos k π + sin k π ; (e) u (k) = ( )k + k ; (f ) u (k) = + + ( ) k k/ Starting with u () =, u () =, u () =, we define the sequence of tri Fibonacci integers u (k) by adding the previous three to get the next one For instance, u () = u () + u () + u () = (a) Write out the next four tri Fibonacci integers (b) Find a third order iterative equation for the n th tri Fibonacci integer (c) Find an explicit formula for the solution, using a computer to approximate the eigenvalues (d) Do they grow faster than the usual Fibonacci numbers? What is their overall rate of growth? (a) u (k+) = u (k+) + u (k+) + u (k) ; (b) u () =, u (5) =, u (6) = 7, u (7) = ; (c) u(k) 8 89 k + Re ( 98 + 57 i ) ( 96 + 669 i ) k = 8 89 k 775 k 88 cos 76 k + 689 sin 76 k Suppose that Fibonacci s rabbits only live for eight years, [8] (a) Write out an iterative equation to describe the rabbit population (b) Write down the first few terms (c) onvert your equation into a first order iterative system using the method of Exercise (d) t what rate does the rabbit population grow? (a) u (k) = u (k ) + u (k ) u (k 8) (b),,,,, 5, 8,,,, 5, 8,,, 9, 59, 857, 6, 67, (c) u (k) = u (k), u (k+),, u (k+7) T satisfies u (k+) = u (k) where the 8 8 coeffiiter 9/9/ 5 c Peter J Olver
cient matrix has s on the superdiagonal, last row (,,,,,,, ) and all other entries (d) The growth rate is given by largest eigenvalue in magnitude: λ = 59, with u (n) 59 n For more details, see [8] 5 Find the general solution to the iterative system u (k+) i = u (k) i + u(k) i+, i =,, n, where we set u (k) = u (k) n+ = for all k Hint: Use Exercise 86 u (k) i = nx j = c j cos j π k sin ij π, i =,, n n + n + 6 Prove that the curves E k = { T k x x = }, k =,,,, sketched in Figure form a family of ellipses with the same principal axes What are the semi-axes? Hint: Use Exercise 85 The key observation is that coefficient matrix T is symmetric Then, according to Exercise 85, the principal axes of the ellipse E = { T x x = } are the orthogonal eigenvectors of T Moreover, T k is also symmetric and has the same eigenvectors Hence, all the ellipses E k have the same principal axes The semi-axes are the absolute values of the eigenvalues, and hence E k has semi-axes (8) k and () k 7 Plot the ellipses E k = { T k x x = } for k =,,, for the following matrices T Then determinetheir principal axes, semi-axes, and areas Hint: Use Exercise 85 (a), (b), (c) 5 5 (a) itz E : principal axes: E : principal axes: E : principal axes: E : principal axes:,,,,, semi-axes:,, area: π, semi-axes:, 9, area: 9 π, semi-axes:, 7, area: 7 π, semi-axes:, 8, area: 8 π (b) itz E : principal axes:,, semi-axes:,, area: 8 π = 58 E : circle of radius 8, area: π = 78 E : principal axes:,, semi-axes: 576, 9, area: 6 π = 7 E : circle of radius, area: 5π = 68 (c) itz 67 7678 E : principal axes:,, semi-axes:, 99, area: π = 566 7678 67 6765 765 E : principal axes:,, semi-axes: 9, 59, area: 6 π = 57 765 6765 69 799 E : principal axes:,, semi-axes: 77, 6, area: 6 π = 799 69 5 5 iter 9/9/ 5 c Peter J Olver
E : principal axes: 78, 7 7, semi-axes: 55,, area: 56 π = 8 78 8 Let T be a positive definite matrix Let E n = { T n x x = }, n =,,,, be the image of the unit circle under the n th power of T (a) Prove that E n is an ellipse True or false: (b) The ellipses E n all have the same principal axes (c) The semi-axes are given by r n = r n, s n = sn (d) The areas are given by n = παn where α = /π (a) See Exercise 85 (b) True they are the eigenvectors of T (c) True r, s are the eigenvalues of T (d) True, since the area is π times the product of the semi-axes, so = πr s, so α = r s = det T Then n = πr n s n = πr n sn = π det T n = πα n 9 nswer Exercise 8 when T an arbitrary nonsingular matrix Hint: Use Exercise 85 (a) Follows from Exercise 85 (b) False; see Exercise 7(c) for a counterexample (c) False the singular values of T n are not, in general, the n th powers of the singular values of T (d) True, since the product of the singular values is the absolute value of the determinant, n = π det T n Given the general solution (9) of the iterative system u (k+) = T u (k), write down the solution to v (k+) = α T v (k) + β v (k), where α, β are fixed scalars v (k) = c (αλ + β) k v + + c n (αλ n + β) k v n Prove directly that if the coefficient matrix of a linear iterative system is real, both the real and imaginary parts of a complex solution are real solutions If u (k) = x (k) + i y (k) is a complex solution, then the iterative equation becomes x (k+) + i y (k+) = T x (k) + i T y (k) Separating the real and imaginary parts of this complex vector equation and using the fact that T is real, we deduce x (k+) = T x (k), y (k+) = T y (k) Therefore, x (k), y (k) are real solutions to the iterative system QED Explain why the solution u (k), k, to the initial value problem (6) exists and is uniquely defined Does this hold if we allow negative k <? The formula uniquely specifies u (k+) once u (k) is known Thus, by induction, once the initial value u () is fixed, there is only one possible solution for k =,,, Existence and uniqueness also hold for k < when T is nonsingular, since u ( k ) = T u ( k) If T is singular, the solution will not exist for k < if any u ( k) rng T, or, if it exists, is not unique since we can add any element of ker T to u ( k) without affecting u ( k+), u ( k+), Prove that if T is a symmetric matrix, then the coefficients in (9) are given by the formula c j = a T v j / v T j v j ccording to Theorem 8, the eigenvectors of T are real and form an orthogonal basis of R n with respect to the Euclidean norm The formula for the coefficients c j thus follows directly from (58) Explain why the j th column c (k) j system c (k+) j of the matrix power T k satisfies the linear iterative = T c (k) j with initial data c () j = e j, the j th standard basis vector Since matrix multiplication acts column-wise, cf (), the j th column of the matrix equation T k+ = T T k is c (k+) j = T c (k) j Moreover, T = I has j th column c () j = e j QED iter 9/9/ 55 c Peter J Olver
5 Let z (k+) = λ z (k) be a complex scalar iterative equation with λ = µ + i ν Show that its real and imaginary parts x (k) = Re z (k), y (k) = Im z (k), satisfy a two-dimensional real linear iterative system Use the eigenvalue method to solve the real system, and verify that your solution coincides with the solution to the original complex equation x (k+) y (k+) with eigenvectors x (k) y (k) = µ ν x (k) ν µ y (k) and so the solution is i = x() + i y () The eigenvalues of the coefficient matrix are µ ± i ν, (µ + i ν) k + x() i y () (µ i ν) k i i Therefore z (k) = x (k) + i y (k) = (x () + i y () )(µ + i ν) k = λ k z () QED 6 Let T be an incomplete matrix, and suppose w,, w j is a Jordan chain associated with an incomplete eigenvalue λ (a) Prove that, for any i =,, j, T k w i = λ k w i + k λ k w i + k λ k w i + () (b) Explain how to use a Jordan basis of T to construct the general solution to the linear iterative system u (k+) = T u (k) (a) Proof by induction: T k+ w i = T λ k w i + k λ k w i + k λ k w i + = λ k T w i + k λ k T w i + k λ k T w i + = λ k (λw i + w i ) + k λ k (λw i + w i ) + k λ k (λw i + w i ) + = λ k+ w i + (k + )λ k w i + k + λ k w i + (b) Each Jordan chain of length j is used to construct j linearly independent solutions by formula () Thus, for an n-dimensional system, the Jordan basis produces the required number of linearly independent (complex) solutions, and the general solution is obtained by taking linear combinations Real solutions of a real iterative system are obtained by using the real and imaginary parts of the Jordan chain solutions corresponding to the complex conjugate pairs of eigenvalues 7 Use the method Exercise 6 to find the general real solution to the following iterative systems: (a) u (k+) = u (k) + v (k), v (k+) = v (k), (b) u (k+) = u (k) + v (k), v (k+) = u (k) + 5v (k), (c) u (k+) = u (k) + v (k) + w (k), v (k+) = v (k) + w (k), w (k+) = w (k), (d) u (k+) = u (k) v (k), v (k+) = u (k) + v (k) + w (k), w (k+) = v (k) + w (k), (e) u (k+) = u (k) v (k) w (k), v (k+) = u (k) +v (k) +w (k), w (k+) = u (k) +v (k) +w (k), (f ) u (k+) = v (k) + z (k), v (k+) = u (k) + w (k), w (k+) = z (k), z (k+) = w (k) (a) u (k) = k c + k c, v (k) = k c ; iter 9/9/ 56 c Peter J Olver
(b) u (k) = kh c + k i c, v (k) = k h c + k c i ; (c) u (k) = ( ) kh c k c + k(k )c i, v (k) = ( ) k h i c (k + )c, w (k) = ( ) k c ; (d) u (k) = kh c + k c + 8 k (k ) + i c, v (k) = k h c + k c i, w (k) = kh c + k c + 8 k (k )c i ; (e) u () = c, v () = c + c, w (k) = c + c, while, for k >, u (k) = k c + k c, v (k) = k c, w (k) = k c + k c ; (f ) u (k) = i k+ c k i k c ( i ) k+ c k( i ) k c, w (k) = i k+ c ( i ) k+ c, v (k) = i k c + k i k c + ( i ) k c + k( i ) k c, z (k) = i k c + ( i ) k c 8 Find a formula for the k th power of a Jordan block matrix Hint: Use Exercise 6 J k λ,n = λ k k λ k k λ k λ k k λ k k k λ k λ k λ k k λ k k n λ k n+ k n λ k n+ k n λ k n+ k n λ k n+ λ k λ n 9 n affine iterative system has the form u (k+) = T u (k) + b, u () = c (a) Under what conditions does the system have an equilibrium solution u (k) u? (b) In such cases, find a formula for the general solution Hint: Look at v (k) = u (k) u (c) Solve the following affine iterative systems: (i) u (k+) 6 = u (k) +, u () =, (ii) u (k+) = u (k) +, u () =, (iii) u (k+) = 6 u (k) +, u () = 6 5 5 6 6 u(k) 6 + (iv) u (k+) =, u() = what happens in cases when there is no fixed point, assuming that T is complete, 6 (d) Discuss (a) The system has an equilibrium solution if and only if (T I )u = b In particular, if is not an eigenvalue of T, every b leads to an equilibrium solution (b) Since v (k+) = T v (k), the general solution is (c) (i) u (k) = u (k) = u + c λ k v + c λk v + + c n λk n v n, where v,, v n are the linearly independent eigenvectors and λ,, λ n the corresponding eigenvalues of T 5 k ( ) k ; iter 9/9/ 57 c Peter J Olver
(ii) u (k) = + ( ) k ( + ) k (iii) u (k) = + 5 ( )k 5( ) k (iv) u (k) = 6 5 + 7 k (d) In general, using induction, the solution is 7 k ; ; + 7 6 k u (k) = T k c + ( I + T + T + + T k )b If we write b = b v + + b n v n, c = c v + + c n v n, in terms of the eigenvectors, then u (k) nx h = λ k j c j + ( + λ j + λ j + + i λk j )b j vj j = If λ j, one can use the geometric sum formula + λ j + λ j + + λk j = λk j λ j, while if λ j =, then + λ j + λ j + + λk j = k Incidentally, the equilibrium solution is u = X b j v λ j j λ j well-known method of generating a sequence of pseudo-random integers x, x, in the interval from to n is based on the Fibonacci equation u (k+) = u (k+) +u (k) mod n, with the initial values u (), u () chosen from the integers,,,, n (a) Generate the sequence of pseudo-random numbers that result from the choices n =, u () =, u () = 7 Keep iterating until the sequence starts repeating (b) Experiment with other sequences of pseudo-random numbers generated by the method listability Stability Determine the spectral radius of the following matrices: (a), (b), (c), (d) 5+ (a) Eigenvalues: 57, 5 7; spectral radius: (b) Eigenvalues: ± i 6 ±785 i ; spectral radius: 6 785 (c) Eigenvalues:,, ; spectral radius: (d) Eigenvalues:, ± i ; spectral radius: 7 Determine whether or not the following matrices are convergent: 5 9 5+ 57 iter 9/9/ 58 c Peter J Olver
(a), (b) 6, (c) 7 5 5, (d) 5 (a) Eigenvalues: ± i ; spectral radius: 656; not convergent (b) Eigenvalues: 95, 586; spectral radius: 95; convergent (c) Eigenvalues: 5, 5, ; spectral radius: 5 ; convergent (d) Eigenvalues:, 57, 7; spectral radius: ; not convergent 8 6 5 Which of the listed coefficient matrices defines a linear iterative system with asymptotically stable zero solution? (a) (e), (b), (f ), (c), (g), (d), 6 (a) Unstable eigenvalues, ; (b) unstable eigenvalues 5+ 7 867, 5 7 95; (c) asymptotically stable eigenvalues ± i ; (d) stable eigenvalues, ± i ; (e) unstable eigenvalues 5,, ; (f ) unstable eigenvalues,, ;, (g) asymptotically stable eigenvalues,, 5 (a) Determine the eigenvalues and spectral radius of the matrix T = (b) Use this information to find the eigenvalues and spectral radius of b T = 5 5 5 5 5 5 5 (c) Write down an asymptotic formula for the solutions to the iterative system u (k+) = b T u (k) (a) λ =, λ = + i, λ = i, ρ(t ) = (b) λ = 5, λ = 5 + 5 i, λ = 5 5 i, ρ(e T ) = 5 (c) u (k) c 5 k (,, ) T, provided the initial data has a non-zero component, c, in the direction of the dominant eigenvector (,, ) T 5 (a) Show that the spectral radius of T = is ρ(t ) = (b) Show that most iterates u (k) = T k u () become unbounded as k (c) Discuss why the inequality u (k) ρ(t ) k does not hold when the coefficient matrix is incomplete (d) an you prove that (8) holds in this example? iter 9/9/ 59 c Peter J Olver
(a) T has a double eigenvalue of, so ρ(t ) = (b) Set u () = a Then T b k = k, and so u (k) = a + k b b provided b (c) In this example, u (k) = b k + abk + a + b bk when b, while ρ(t ) k = is constant, so eventually u (k) > ρ(t ) k no matter how large is (d) For any σ >, we have bk σ k for k provided is sufficiently large more specifically, if > b/(e log σ); see Exercise 6 Given a linear iterative system with non-convergent matrix, which solutions, if any, will converge to? solution u (k) if and only if the initial vector u () = c v + + c j v j is a linear combination of the eigenvectors (or more generally, Jordan chain vectors) corresponding to eigenvalues satisfying λ i <, i =,, j 7 Prove that if is any square matrix, then there exists c such that the scalar multiple c is a convergent matrix Find a formula for the largest possible such c Since ρ(c) = c ρ(), then c is convergent if and only if c < /ρ() So, technically, there isn t a largest c 8 Suppose T is a complete matrix (a) Prove that every solution to the corresponding linear iterative system is bounded if and only if ρ(t ) (b) an you generalize this result to incomplete matrices? Hint: Look at Exercise 6 (a) Let u,, u n be a unit eigenvector basis for T, so u j = Let m j = max n c j c u + + c n u n o, which is finite since we are maximizing a continuous function over a closed, bounded set Set m = max{m,, m n } Now, if u () = c u + + c n u n < ε, then c j < m ε for j =,, n Therefore, by (5), u (k) = c + + c n n m ε, and hence the solution remains close to QED (b) If any eigenvalue of modulus λ = is incomplete, then, according to (), the system has solutions of the form u (k) = λ k w i + k λ k w i +, which are unbounded as k Thus, the origin is not stable in this case On the other hand, if all avs of modulus are complete, then the system is stable, even if there are incomplete eigenvalues of modulus < 9 Suppose a convergent iterative system has a single dominant real eigenvalue λ Discuss how the asymptotic behavior of the real solutions depends on the sign of λ ssume u () = c v + + c n v n with c For k, u (k) c λ k v since λ k λk j for all j > Thus, the entries satisfy u(k+) i λ u (k) i and so, if nonzero, are just multiplied by λ Thus, if λ > we expect to see the signs of all the entries of u (k) not change for k sufficiently large, whereas if λ <, the signs alterate at each step of the iteration (a) Discuss the asymptotic behavior of solutions to a convergent iterative system that has two eigenvalues of largest modulus, eg, λ = λ How can you detect this? (b) Discuss the case when is a real matrix with a complex conjugate pair of dominant eigenvalues iter 9/9/ 55 c Peter J Olver
Writing u () = c v + + c n v n, then for k, u (k) λ k (c v + c (λ /λ )k v ) T Suppose T has spectral radius ρ(t ) an you predict the spectral radius of ct + d I, where c, d are scalars? If not, what additional information do you need? If T has eigenvalues λ j, then ct + d I has eigenvalues cλ j + d However, it is not necessarily true that the dominant eigenvalue of ct + d I is cλ + d when λ is the dominant eigenvalue of T For instance, if λ =, λ =, so ρ(t ) =, then λ =, λ =, so ρ(t I ) = ρ(t ) Thus, you need to know all the eigenvalues to predict ρ(t ) (In more detail, it actually suffices to know the extreme eigenvalues, ie, those such that the other eigenvalues lie in their convex hull in the complex plane) Let have singular values σ σ n Prove that T is a convergent matrix if and only if σ < (Later we will show that this implies that itself is convergent) y definition, the eigenvalues of T are λ i = σ i, and so the spectral radius of T is equal to ρ( T ) = max{σ,, σ n } Thus ρ(t ) = λ < if and only if σ = q λ < Let M n be the n n tridiagonal matrix with all s on the sub- and super-diagonals, and zeros on the main diagonal (a) What is the spectral radius of M n? Hint: Use Exercise 86 (b) Is M n convergent? (c) Find the general solution to the iterative system u (k+) = M n u (k) (a) ρ(m n ) = cos nx c j j = cos j π n + π n+ (b) No since its spectral radius is slightly less than (c) u(k) i = k sin ij π, i =,, n n + Let α, β be scalars Let T α,β be the n n tridiagonal matrix that has all α s on the sub- and super-diagonals, and β s on the main diagonal (a) Solve the iterative system u (k+) = T α,β u (k) (b) For which values of α, β is the system asymptotically stable? Hint: ombine Exercises and (a) u (k) nx j π k i = c j β + α cos sin ij π, i =,, n j = n + n + π (b) Stable if and only if ρ(t α,β ) = ( β ± α cos < In particular, if β ± α < n + the system is asymptotically stable for any n 5 (a) Prove that if det T > then the iterative system u (k+) = T u (k) is unstable (b) If det T < is the system necessarily asymptotically stable? Prove or give a counterexample (a) ccording to Exercise 8, T has at least one eigenvalue with λ > (b) No For example T = has det T = but ρ(t ) = 6 True or false: (a) ρ(c) = c ρ(), (b) ρ(s S) = ρ(), (c) ρ( ) = ρ(), (d) ρ( ) = /ρ(), (e) ρ( + ) = ρ() + ρ(), (f ) ρ() = ρ() ρ() (a) False: ρ(c ) = c ρ() (b) True (c) True (d) False since ρ() = max λ whereas ρ( ) = max /λ (e) False in almost all cases (f ) False iter 9/9/ 55 c Peter J Olver
7 True or false: (a) If is convergent, then is convergent (b) If is convergent, then T is convergent (a) True by part (c) of Exercise 6 (b) False = whereas T = 5 has ρ( T ) = + = 57 has ρ() = 8 True or false: If the zero solution of the differential equation u = u is asymptotically stable, so is the zero solution of the iterative system u (k+) = u (k) False The first requires Re λ j < ; the second requires λ j < 9 Suppose T k as k (a) Prove that = (b) an you characterize all such matrices? (c) What are the conditions on the matrix T for this to happen? (a) = lim T k «= lim T k = (b) The only eigenvalues of are and k k Moreover, must be complete, since if v, v are the first two vectors in a Jordan chain, then v = λ v, v = λ v + v, with λ = or, but v = λ v + λv v = λ v + v, so there are no Jordan chains except for the ordinary eigenvectors Therefore, = S diag (,,,, ) S for some nonsingular matrix S (c) If λ is an eigenvalue of T, then either λ <, or λ = and is a complete eigenvalue Prove that a matrix with all integer entries is convergent if and only if it is nilpotent, ie, k = O for some k Give a nonzero example of such a matrix If v has integer entries, so does k v for any k, and so the only way in which k v is if k v = for some k Now consider the basis vectors e,, e n Let k i be such that k i e i = Let k = max{k,, k n }, so k e i = for all i =,, n Then k I = k = O, and hence is nilpotent QED The simplest example is onsider a second order iterative scheme u (k+) = u (k+) + u (k) Define a generalized eigenvalue to be a complex number that satisfies det(λ I λ ) = Prove that the system is asymptotically stable if and only if all its generalized eigenvalues satisfy λ < Hint: Look at the equivalent first order system and use Exercise 9b The equivalent first order system v (k+) = v (k) for v (k) = u (k) u (k+) has coefficient matrix = O I To compute the eigenvalues of we form det( λ I ) = λ I I det Now use row operations to subtract appropriate multiples of the first λ I n rows from the last n, and the a series of row interchanges to conclude that det( λ I ) = det λ I + λ λ I = ± det( + λ λ I ) I O = ± det + λ λ I λ I Thus, the generalized eigenvalues are the same as the eigenvalues of, and hence stability requires they all satisfy λ < O I iter 9/9/ 55 c Peter J Olver
Let p(t) be a polynomial ssume < λ < µ Prove that there is a positive constant such that p(n) λ n < µ n for all n > Set σ = µ/λ > If p(x) = c k x k + + c x + c has degree k, then p(n) an k for all n where a = max c i To prove an k σ n it suffice to prove that k log n < n log σ + log log a Now h(n) = n log σ k log n has a minimum when h (n) log σ k/n =, so n = k/ log σ The minimum value is h(k/ log σ) = k( log(k/ log σ)) Thus, choosing log > log a + k(log(k/ log σ) ) will ensure the desired inequality QED Prove the inequality (8) when T is incomplete Use it to complete the proof of Theorem in the incomplete case Hint: Use Exercises 6, ccording to Exercise 6, there is a polynomial p(x) such that u (k) X i λ i k p i (k) p(k) ρ() k Thus, by Exercise, u (k) σ k for any σ > ρ() QED Suppose that M is a nonsingular matrix (a) Prove that the implicit iterative scheme M u (n+) = u (n) is asymptotically stable if and only if all the eigenvalues of M are strictly greater than one in magnitude: µ i > (b) Let K be another matrix Prove that iterative scheme M u (n+) = K u (n) is asymptotically stable if and only if all the generalized eigenvalues of the matrix pair K, M, as in Exercise 88 are strictly less than one in magnitude: λ i < (a) Rewriting the system as u (n+) = M u (n), stability requires ρ(m ) < The eigenvalues of M are the reciprocals of the eigenvalues of M, and hence ρ(m ) < if and only if / µ i < QED (b) Rewriting the system as u (n+) = M K u (n), stability requires ρ(m K) < Moreover, the eigenvalues of M K coincide with the generalized eigenvalues of the pair; see Exercise 95 for details 5 Find all fixed points for the linear iterative systems with the following coefficient matrices: (a), (b), (c), (d) 8 7 7 6 (a) all scalar multiples of (d) all linear combinations of, ; (b) ; (c) all scalar multiples of ; 6 (a) Discuss the stability of each fixed point and the asymptotic behavior(s) of the solutions to the systems in Exercise 5 (b) Which fixed point, if any, does the solution with initial condition u () = e converge to? (a) The eigenvalues are,, so the fixed points are stable, while all other solutions go to a unique fixed point at rate k When u () = (, ) T, then u (k) 5, 5 T iter 9/9/ 55 c Peter J Olver
(b) The eigenvalues are 9, 8, so the origin is a stable fixed point, and every nonzero solution goes to it, most at a rate of 9 k When u () = (, ) T, then u (k) also (c) The eigenvalues are,,, so the fixed points are unstable Most solutions, specifically those with a nonzero component in the dominant eigenvector direction, become unbounded However, when u () = (,, ) T, then u (k) = (,, ) T for k, and the solution stays at a fixed point (d) The eigenvalues are 5 and, so the fixed points are unstable Most solutions, specifically those with a nonzero component in the dominant eigenvector direction, become unbounded, including that with u () = (,, ) T 7 Suppose T is a symmetric matrix that satisfies the hypotheses of Theorem 7 with a simple eigenvalue λ = Prove the solution to the linear iterative system has limiting value lim k u(k) = u() v v v Since T is symmetric, its eigenvectors v,, v n form an orthonormal basis of R n Writing u () = c v + + c n v n, the coefficients are given by the usual orthogonality formula (57): c i = u () v i / v Moreover, since λ =, while λ j < for j, u (k) = c v + c λ k v + + c n λk n v n c v = u() v v v QED 8 True or false: If T has a stable nonzero fixed point, then it is a convergent matrix False T has an eigenvalue of, but convergence requires all eigenvalues less than in modulus 9 True or false: If every point u R n is a fixed point, then they are all stable haracterize such systems True In this case T = I and all solutions remain fixed (a) Under what conditions does the linear iterative system u (k+) = T u (k) have a period solution, ie, u (k+) = u (k) u (k+)? Give an example of such a system (b) Under what conditions is there a unique period solution? (c) What about a period m solution? (a) The condition u (k+) = T u (k) = u (k) implies that u (k) is an eigenvector of T with eigenvalue of Thus, u (k) is an eigenvector of T with eigenvalue ; if the eigenvalue were then u (k) = u (k+), contrary to assumption Thus, the iterative system has a period solution if and only if T has an eigenvalue of (b) T = has the period orbit u (k) = c( )k for any c (c) The period solution is never unique since any nonzero scalar multiple is also a period solution (d) T must have an eigenvalue equal to a primitive m th root of unity Prove Theorem 8, (a) assuming T is complete; (b) for general T Hint: Use Exercise 6 (a) iter 9/9/ 55 c Peter J Olver
mnorm Matrix Norms and the Gerschgorin Theorem ompute the matrix norm of the following matrices Which are guaranteed to be convergent? (e) (a) 7 7 7 6 7 7 7 7 7 6, (f ), (b) 5 7 6 5 6 8, (g) 8, (c) 7 7 6 7 7, (d) f r f r, 5, (a), convergent (b), inconclusive (c) 8 7, inconclusive (d) 7, inconclusive (e) 7 8, inconclusive (f ) 9, convergent (g) 7, inconclusive (h), inconclusive ompute the Euclidean matrix norm of each matrix in Exercise Have your convergence conclusions changed? (a) 67855, convergent (b) 57, inconclusive (c) 9755, convergent (d) 957, inconclusive (e) 66, inconclusive (f ) 8, convergent (g) 6, inconclusive (h) 769, convergent ompute the spectral radii of the matrices in Exercise Which are convergent? ompare your conclusions with those of Exercises and (a), convergent (b), convergent (c) 9755, convergent (d) 8, divergent (e) 97, convergent (f ) 8, convergent (g), convergent (h), convergent k Let k be an integer and set k = k ompute (a) k k, (b) k, (c) ρ( k ) (d) Explain why every k is a convergent matrix, even though their matrix norms can be arbitrarily large (e) Why does this not contradict orollary? (a) k = k + k, (b) k = k +, (c) ρ( k ) = (d) Thus, a convergent matrix can have arbitrarily large norm (e) ecause the norm will depend on k 5 Find a matrix such that (a) ; (b), so =, then When = (a) =, = (b) = r + 5 = 68, = 6 Show that if c < / then c is a convergent matrix Since c = c < q + = 7 Prove that the spectral radius function does not satisfy the triangle inequality by finding matrices, such that ρ( + ) > ρ() + ρ() =, =, then ρ( + ) = > + = ρ() + ρ() 8 True or false: If all singular values of satisfy σ i < then is convergent iter 9/9/ 555 c Peter J Olver
True this implies = max σ i < 9 Find a convergent matrix that has dominant singular value σ > If = 7 and σ =, then ρ() = The singular values of are σ = = 7 + = True or false: If = S S are similar matrices, then (a) =, (b) =, (c) ρ() = ρ() (a) False: if =, S =, then = S S = and = = ; (b) false: same example has = 5 = ; (c) true, since and have the same eigenvalues Prove that the condition number of a nonsingular matrix is given by κ() = y definition, κ() = σ /σ n Now = σ On the other hand, by Exercise 85, the singular values of are the reciprocals /σ i of the singular values of, and so the largest one is = /σ n QED (i) Find an explicit formula for the matrix norm (ii) ompute the matrix norm of the matrices in Exercise, and discuss convergence (i) The matrix norm is the maximum absolute column sum (ii) (a) 6 5, convergent (b) 7 6, inconclusive (c) 8 7, inconclusive (d), inconclusive (e) 7, inconclusive (f ) 9, convergent (g) 7, inconclusive (h), convergent Prove directly from the axioms of Definition that () defines a norm on the space of n n matrices If a,, a n are the rows of, then the formula () can be rewritten as = max{ a i }, ie, the maximal norm of the rows Thus, by the properties of the -norm, + = max{ a i + b i } max{ a i + b i } max{ a i } + max{ b i } = +, c = max{ ca i } = max{ c a i } = c max{ a i } = c Finally, since we are maximizing non-negative quantities; moreover, = if and only if all its rows have a i = and hence all a i =, which means = O QED Let K > be a positive definite matrix haracterize the matrix norm induced by the inner product x, y = x T K y Hint: Use Exercise 8 = max{σ,, σ n } is the largest generalized singular value, meaning σ i = q λ i where λ,, λ n are the generalized eigenvalues of the positive definite matrix pair T K and K, satisfying T K v i = λk v i, or, equivalently, the eigenvalues of K T K 5 Let = ompute the matrix norm using the following norms in R : (a) the weighted norm v = max{ v, v }; (b) the weighted norm v = v + v ; (c) the weighted inner product norm v = q v + v ; (d) the norm iter 9/9/ 556 c Peter J Olver
associated with the positive definite matrix K = (a) 6 The Frobenius norm of an n n matrix is defined as F = v nx u t i,j = a ij Prove that this defines a matrix norm by checking the three norm axioms plus the multiplicative inequality () If we identify an n n matrix with a vector in R n, then the Frobenius norm is the same as the ordinary Euclidean norm, and so the norm axioms are immediate To check the multiplicative property, let r T,, rt n denote the rows of and c,, c n the columns of, so F = v u nx t i = F = r i, F = v u nx t i,j = v nx u t j = c ij = v uuut n X i,j = c j Then, setting =, we have (r T i c j ) where we used auchy Schwarz for the inequality v u nx t i,j = r i c j = F F, 7 Let be an n n matrix with singular value vector σ = (σ,, σ r ) Prove that (a) σ = ; (b) σ = F, the Frobenius norm of Exercise 6 Remark: σ also defines a useful matrix norm, known as the Ky Fan norm (a) This is a restatement of Proposition 8 (b) σ = n X i = σ i = n X i = λ i = tr( T ) = nx i,j = a ij = F QED 8 Explain why = max a ij defines a norm on the space of n n matrices Show by example that this is not a matrix norm, ie, () is not necessarily valid If we identify a matrix with a vector in R n, then this agrees with the norm on R n and hence satisfies the norm axioms When =, then =, and so = > = 9 Prove that the closed curve parametrized in (7) is an ellipse What are its semiaxes? If V is a finite-dimensional normed vector space, then it can be proved, [6], that a series X n = v i converges to v V if and only if the series of norms converges: Use this fact to prove that the exponential matrix series (95) converges t n X n = v i < e t X X t n = n = n n and the series of norms is n = e t is the standard n = n scalar exponential series, which, by the ratio or root tests, [], converges for all t QED (a) Use Exercise to prove that the geometric matrix series X n = n converges iter 9/9/ 557 c Peter J Olver
whenever ρ() < Hint: pply orollary (b) Prove that the sum is ( I ) How do you know I is invertible? (a) hoosing a matrix norm such that a = <, the norm series is bounded by a convergent geometric series: X n X n X = a n = a = Therefore, the matrix series converges ( I ) X n = = n = (b) Moreover, n = X n = n X n = n+ = I, since all other terms cancel I is invertible if and only if is not an eigenvalue of, and we are assuming all eigenvalues are less than in magnitude For each of the following matrices (i) Find all Gerschgorin disks; (ii) plot the Gerschgorin domain in the complex plane; (iii) compute the eigenvalues and confirm the truth of the ircle Theorem (a) (e), (b) 7, (f ) 6, (c), 6, (d) (g), (h) (a) Gerschgorin disks: z ; eigenvalues:, (b) Gerschgorin disks: z, z + 6 ; eigenvalues:,, (c) Gerschgorin disks: z, z ; eigenvalues: ± i (d) Gerschgorin disks: z, z ; eigenvalues:,, (e) Gerschgorin disks: z + 6, z 9, z + ; eigenvalues:, ± i 6 (f ) Gerschgorin disks: z =, z, z 5 ; eigenvalues:, ± (g) Gerschgorin disks: z, z ; eigenvalues:, ± i (h) Gerschgorin disks: z, z, z, z ; eigenvalues: ± 5, 5 ± 5 True or false: The Gerschgorin domain of the transpose of a matrix T is the same as the Gerschgorin domain of the matrix, that is D T = D False lmost any non-symmetric matrix, eg, provides a counterexample (i) Explain why the eigenvalues of must lie in its refined Gerschgorin domain D = D T D (ii) Find the refined Gerschgorin domains for each of the matrices in Exercise and confirm the result in part (i) (i) ecause and its transpose T have the same eigenvalues, which must therefore belong to both D and D T (ii) iter 9/9/ 558 c Peter J Olver
(a) Gerschgorin disks: z ; eigenvalues:, (b) Gerschgorin disks: z, z + 6 ; eigenvalues: (c) Gerschgorin disks: z, z ; eigenvalues: ± i (d) Gerschgorin disks: z, z ; eigenvalues:,, (e) Gerschgorin disks: z +, z 6, z + ; eigenvalues:, ± i 6 (f ) Gerschgorin disks: z, z 6, z ; eigenvalues:, ± (g) Gerschgorin disks: z =, z, z ; eigenvalues:, ± i (h) Gerschgorin disks: z, z, z, z ; eigenvalues: ± 5, 5 ± 5 5 Let be a square matrix Prove that max{, t} ρ() s, where s = max{s,, s n } is the maximal absolute row sum of, as defined in (9), and t = min n a ii r i o, with r i given by () y elementary geometry, all points z in a closed disk of radius r centered at z = a satisfy max{, a r } z a + r Thus, every point in the i th Gerschgorin disk satisfies max{, a ii r i } z a ii + r i = s i Since every eigenvalue lies in such a disk, they all satisfy max{, t} λ i s, and hence ρ() = max{ λ i } does too QED 6 (a) Suppose that every entry of the n n matrix is bounded by a ij < n Prove that is a convergent matrix Hint: Use Exercise 5 n n with one or more entries satisfying a ij = n (a) The absolute row sums of are bounded by s i = (b) = max s i < by Exercise 5, (b) Produce a matrix of size that is not convergent nx j = has eigenvalues, and hence ρ() = a ij <, and so ρ() s = 7 Suppose the largest entry (in modulus) of is a ij = a How large can its radius of convergence be? Using Exercise 5, we find ρ() s = max n s i o n a 8 Write down an example of a diagonally dominant matrix that is also convergent ny diagonal matrix whose diagonal entries satisfy < a ii < 9 True or false: (a) positive definite matrix is diagonally dominant (b) diagonally dominant matrix is positive definite oth false is a counterexample to (a), while is a counterexample 5 to (b) However, see Exercise Prove that if K is symmetric, diagonally dominant, and each diagonal entry is positive, then K is positive definite The eigenvalues of K are real by Theorem 8 The i th Gerschgorin disk is centered at k ii > and by diagonal dominance its radius is less than the distance from its center to the origin Therefore, all eigenvalues of K must be positive and hence, by Theorem 8, K > iter 9/9/ 559 c Peter J Olver
(a) Write down an invertible matrix whose Gerschgorin domain contains (b) an you find an example which is also diagonally dominant? (a) = Proposition 7 has Gerschgorin domain z (b) No see the proof of Prove that if is diagonally dominant and each diagonal entry is negative, then the zero equilibrium solution to the linear system of ordinary differential equations u = u is asymptotically stable The i th Gerschgorin disk is centered at a ii < and, by diagonal dominance, its radius is less than the distance to the origin Therefore, all eigenvalues of lie in the left half plane: Re λ <, which, by Theorem 95, implies asymptotic stability of the differential equation MP Markov Processes Determine if the following matrices are regular transition matrices If so, find the associated probability eigenvector (e) (j), (f ) 5 8, (k) 5 7 (a) Not a transition matrix; (b) not a transition matrix; (a) 5 5, (g) (c) regular transition matrix: 8 7, 9 (d) regular transition matrix: 6, 5 6 (e) not a regular transition matrix;, (b) 5, (l) T 7 ; T ; T ; 5, (h) 8, (c), (d) 5, (i) 7 6 5 5, (m) 5 7 (f ) regular transition matrix:,, (g) regular transition matrix: ( 5, 8, 7 ) T ; (h) not a transition matrix; (i) regular transition matrix: 6,, T = ( 65, 77, 8 ) T ; (j) not a regular transition matrix; (k) not a transition matrix; (l) regular transition matrix ( has all positive entries): 5, 5, 5, 9 ( 579, 775, 765, 897 ) T ; (m) regular transition matrix: ( 59, 9, 977, 6 ) T 5 5, 7 8 5 8 T = iter 9/9/ 56 c Peter J Olver
study has determined that, on average, the occupation of a boy depends on that of his father If the father is a farmer, there is a % chance that the son will be a blue collar laborer, a % chance he will be a white collar professional, and a % chance he will also be a farmer If the father is a laborer, there is a % chance that the son will also be one, a 6% chance he will be a professional, and a % chance he will be a farmer If the father is a professional, there is a 7% chance that the son will also be one, a 5% chance he will be a laborer, and a 5% chance he will be a farmer (a) What is the probability that the grandson of a farmer will also be a farmer? (b) In the long run, what proportion of the male population will be farmers? (a) 5%; (b) 976% farmers, 68% laborers, 6% professionals The population of an island is divided into city and country residents Each year, 5% of the residents of the city move to the country and 5% of the residents of the country move to the city In, 5, people live in the city and 5, in the country ssuming no growth in the population, how many people will live in the city and how many will live in the country between the years and 8? What is the eventual population distribution of the island? : 7, city,, country 5: 8,6 city,, country 6: 9,88 city,, country 7:,9 city, 996 country 8:,7 city, 8,77 country Eventual: 5, in the city and 5, in the country student has the habit that if she doesn t study one night, she is 7% certain of studying the next night Furthermore, the probability that she studies two nights in a row is 5% How often does she study in the long run? 58% of the nights 5 travelingsalesman visits the three cities of tlanta, oston, and hicago The matrix 5 describes the transition probabilities of his trips Describe his travels in 5 5 5 words, and calculate how often he visits each city on average When in tlanta he always goes to oston; when in oston he has a 5% probability of going to either tlanta or hicago; when in hicago he has a 5% probability of going to either tlanta or oston Note that the transition matrix is regular because has all positive entries In the long run, his visits average: tlanta: %, oston: %, hicago: % 6 business executive is managing three branches, labeled, and, of a corporation She never visits the same branch on consecutive days If she visits branch one day, she visits branch the next day If she visits either branch or that day, then the next day she is twice as likely to visit branch as to visit branch or Explain why the resulting transition matrix is regular Which branch does she visit the most often in the long run? The transition matrix is 5% and branch : 5% She visits branch % of the time, branch 7 certain plant species has either red, pink or white flowers, depending on its genotype If you cross a pink plant with any other plant, the probabilitydistribution of the 5 5 offspring are prescribed by the transition matrix T = 5 5 5 On average, if you 5 5 iter 9/9/ 56 c Peter J Olver