Linear Algebra

406 Chapter Five. Similarityn 1 2 3 4 5 6 7 8 9 10T (n) 1 3 7 15 31 63 127 255 511 1023We recognize those as being simply one less than a power of two.To derive this equation instead of just guessing at it, we write the originalrelation as −1 = −T (n + 1) + 2T (n), consider the homogeneous relation 0 =−T (n) + 2T (n − 1), get its associated polynomial −λ + 2, which obviously hasthe single, unique, root of r 1 = 2, and conclude that functions satisfying thehomogeneous relation take the form T (n) = c 1 2 n .That’s the homogeneous solution. Now we need a particular solution.Because the nonhomogeneous relation −1 = −T (n + 1) + 2T (n) is so simple,in a few minutes (or by remembering the table) we can spot the particularsolution T (n) = −1 (there are other particular solutions, but this one is easilyspotted). So we have that — without yet considering the initial condition — anysolution of T (n + 1) = 2T (n) + 1 is the sum of the homogeneous solution andthis particular solution: T (n) = c 1 2 n − 1.The initial condition T (1) = 1 now gives that c 1 = 1, and we’ve gotten theformula that generates the table: the n-disk Tower of Hanoi problem requires aminimum of 2 n − 1 moves.Finding a particular solution in more complicated cases is, naturally, morecomplicated. A delightful and rewarding, but challenging, source on recurrencerelations is [Graham, Knuth, Patashnik]., For more on the Tower of Hanoi,[Ball & Coxeter] or [Gardner 1957] are good starting points. So is [Hofstadter].Some computer code for trying some recurrence relations follows the exercises.Exercises1 Solve each homogeneous linear recurrence relations.(a) f(n + 1) = 5f(n) − 6f(n − 1)(b) f(n + 1) = 4f(n − 1)(c) f(n + 1) = 6f(n) + 7f(n − 1) + 6f(n − 2)2 Give a formula for the relations of the prior exercise, with these initial conditions.(a) f(0) = 1, f(1) = 1(b) f(0) = 0, f(1) = 1(c) f(0) = 1, f(1) = 1, f(2) = 3.3 Check that the isomorphism given betwween S and R k is a linear map. It isargued above that this map is one-to-one. What is its inverse?4 Show that the characteristic equation of the matrix is as stated, that is, is thepolynomial associated with the relation. (Hint: expanding down the final column,and using induction will work.)5 Given a homogeneous linear recurrence relation f(n + 1) = a nf(n) + · · · +a n−k f(n − k), let r 1 , . . . , r k be the roots of the associated polynomial.(a) Prove that each function f ri (n) = r n k satisfies the recurrence (without initialconditions).(b) Prove that no r i is 0.(c) Prove that the set {f r1 , . . . , f rk } is linearly independent.