(b) Find the constituent matrices of A. For this, we need the eigenvalues of A, which we can find by using the Maple command "eigenvals":

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Brittney Wilkins
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1 Problem 5 : The card deal problem First solution: using the linalg package (a) Defining the matrix A. First define B = 13A: (1) (b) Find the constituent matrices of A. For this, we need the eigenvalues of A, which we can find by using the Maple command "eigenvals": Unfortunately, Maple lists these in no particular order. Thus, a better method is to observe that these are just the diagonal entries of A because A is lower triangular. In other words, the eigenvalues are given by: The characteristic polynomial of A is therefore: (2) (3) (4) Since roots of ch(t) are all distinct, we can compute the i-th constituent polynomial of ch(t) by the following formula:

2 Note that we first have to simplify the expression ch/(t-ev[i]) before we can evaluate it at t =ev[i]. The first constituent polynomial e_10 is therefore: (5) The constituent matrices of A are therefore obtained by: For example, the first 3 constituent matrices are: (6)

3 (7) Since the pattern is now clear, we only compute but do not display the rest. For this, is convenient to put these into a list: Note, however, that the last matrix E90 does not fit into the pattern of the others: (8) (c) Find a formula for A^n and calculate p_n. By the spectral decomposition formula, A^n is given by: Check this: true (9) (10)

4 (11) As is explained in the text, the probability distribution u_n of the states after n+1 deals is given by the formula u_n = A^n*u_0, u_0 = (1/13)(1,1,..,1,5). Since the state s_6 represents the state that 7 is the highest card, we are interested in the probability p_n of state s_6 after n deals. This is given by p_n = e_6*u_(n-1) = e_6*a^(n-1)*u_0, where e_6 = (0,0,0,0,0,0,1,0,0,0). (Note the shift in the index/exponent.) (12) (d) Calculate p_n explicitly for n \le 10. (13) (14) Note that the first three values agree with those given in the book. Their numerical equivalents are:

5 (15) ************************************************************************* Second solution: using the LinearAlgebra package (a) Defining the matrix A. First define B = 13A: (16) (b) Find the constituent matrices of A. For this, we need the eigenvalues of A, which we can find by using the Maple command "Eigenvalues": (17)

6 (17) Unfortunately, Maple does not list these in increasing order. Thus, a better method is to observe that these are just the diagonal entries of A because A is lower triangular. In other words, the eigenvalues are given by: The characteristic polynomial of A is therefore: (18) (19) Since roots of ch(t) are all distinct, we can compute the i-th constituent polynomial of ch(t) by the following formula: Note that we first have to simplify the expression ch/(t-ev[i]) before we can evaluate it at t =ev[i]. The first constituent polynomial e_10 is therefore: (20)

7 The constituent matrices of A are therefore obtained by: Note: The command "subs(t=a, ei0(i))" by itself does not evaluate this matrix polynomial correctly. Since there dos not seem to an equvialent of the linalg command "evalm" in the LinearAlgebra package, we use evalm and then convert the matrix to a Matrix. Another method that seems to work is to first expand the polynomial: For example, the first 3 constituent matrices are: (21) (22) (23)

8 (23) Since the pattern is now clear, we only compute but do not display the rest. For this, is convenient to put these into a list: Note, however, that the last matrix E_90 does not fit into the pattern of the others: (24) (c) Find a formula for A^n and calculate p_n. By the spectral decomposition formula, A^n is given by: Check this: true (25) (26)

9 (27) As is explained in the text, the probability distribution u_n of the states after n+1 deals is given by the formula u_n = A^n*u_0, u_0 = (1/13)(1,1,..,1,5). Since the state s_6 represents the state that 7 is the highest card, we are interested in the probability p_n of state s_6 after n deals. This is given by p_n = e_6*u_(n-1) = e_6*a^(n-1)*u_0, where e_6 = (0,0,0,0,0,0,1,0,0,0). (Note the shift in the index/exponent.)

10 (28) Recall the the LinearAlgebra package distinguishes between row vectors and column vectors! (d) Calculate p_n explicitly for n \le 10. (29) (30) Note that the first three values agree with those given in the textbook. Their numerical equivalents are: (31)

EXAM. Exam 1. Math 5316, Fall December 2, 2012

EXAM. Exam 1. Math 5316, Fall December 2, 2012 EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.