8. The matrix difference equation p(t + 1) = Mp(t), t = 0, 1,2,.. with the column vector (vector of probabilities) p(t) = (p1(t), P2(t),...,Pn(t))" and the n x n matrix (1 – w) 0.5w 0.5w 0.5w (1 – w) 0.5w 0.5w M = (1 – w) 0.5w (1 – w) plays a role in random walk in one dimension. M is called the transition proba- bility matrix and w denotes the probability w E [0, 1] that at a given time step the particle jumps to either of its nearest neighbor sites and the probability that the particle does not jump either to the right of left is (1 – w). The matrix M is of the type known as circulant matriz. Such an n x n matrix is of the form Co C2 ... Cn-1 Co C =| Cn-2 Cn-1 Cn-2 Cn-3 Cn-1 CO C1 C2 C3 CO with the normalized eigenvectors 1 1 e2rij/n , j = 1,...,n. e2(n-1)wij/n (i) Use this result to find the eigenvalues of the matrix C. (ii) Use (i) to find the eigenvalues of the matrix M. (iii) Use (ii) to find p(t) (t = 0,1,2,.), where we expand the initial distribution vector p(0) in terms of the eigenvectors p(0) = arek with EP;(0) = 1. k=1 j=1 (iv) Assume that p(0) = ! (1 1 1)". Give the time evolution of p(0).
8. The matrix difference equation p(t + 1) = Mp(t), t = 0, 1,2,.. with the column vector (vector of probabilities) p(t) = (p1(t), P2(t),...,Pn(t))" and the n x n matrix (1 – w) 0.5w 0.5w 0.5w (1 – w) 0.5w 0.5w M = (1 – w) 0.5w (1 – w) plays a role in random walk in one dimension. M is called the transition proba- bility matrix and w denotes the probability w E [0, 1] that at a given time step the particle jumps to either of its nearest neighbor sites and the probability that the particle does not jump either to the right of left is (1 – w). The matrix M is of the type known as circulant matriz. Such an n x n matrix is of the form Co C2 ... Cn-1 Co C =| Cn-2 Cn-1 Cn-2 Cn-3 Cn-1 CO C1 C2 C3 CO with the normalized eigenvectors 1 1 e2rij/n , j = 1,...,n. e2(n-1)wij/n (i) Use this result to find the eigenvalues of the matrix C. (ii) Use (i) to find the eigenvalues of the matrix M. (iii) Use (ii) to find p(t) (t = 0,1,2,.), where we expand the initial distribution vector p(0) in terms of the eigenvectors p(0) = arek with EP;(0) = 1. k=1 j=1 (iv) Assume that p(0) = ! (1 1 1)". Give the time evolution of p(0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![8. The matrix difference equation
P(t + 1) %3D Мp(),
t = 0, 1, 2, ..
with the column vector (vector of probabilities)
p(t) = (p1(t), p2(t), ...,Pn(t))"
and the n x n matrix
(1 – w)
0.5w
0.5w
0.5w
(1 – w)
0.5w
0.5w
M =
(1 – w)
0.5w
(1 – w).
plays a role in random walk in one dimension. M is called the transition proba-
bility matrix and w denotes the probability w E [0, 1] that at a given time step
the particle jumps to either of its nearest neighbor sites and the probability that
the particle does not jump either to the right of left is (1 – w). The matrix M
is of the type known as circulant matrix. Such an n x n matrix is of the form
Co
C2
Cn-1
Сп-1
CO
Cn-2
C =
Сп-2 Сп-1
CO
Cn-3
C1
C2
C3
CO
with the normalized eigenvectors
1
1
e2mij/n
j = 1,...,n.
e2(n-1)xij/n
(i) Use this result to find the eigenvalues of the matrix C.
(ii) Use (i) to find the eigenvalues of the matrix M.
(iii) Use (ii) to find p(t) (t = 0,1, 2,...), where we expand the initial distribution
vector p(0) in terms of the eigenvectors
p(0) = arek with EP;(0):
= 1.
k=1
j=1
(iv) Assume that p(0) = (1 1
1)". Give the time evolution of p(0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52c3c0a3-7c30-45b2-8671-df30f019d52d%2F1c158a0a-e430-489e-a025-1f9b0cab53d6%2Fzspjpn9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8. The matrix difference equation
P(t + 1) %3D Мp(),
t = 0, 1, 2, ..
with the column vector (vector of probabilities)
p(t) = (p1(t), p2(t), ...,Pn(t))"
and the n x n matrix
(1 – w)
0.5w
0.5w
0.5w
(1 – w)
0.5w
0.5w
M =
(1 – w)
0.5w
(1 – w).
plays a role in random walk in one dimension. M is called the transition proba-
bility matrix and w denotes the probability w E [0, 1] that at a given time step
the particle jumps to either of its nearest neighbor sites and the probability that
the particle does not jump either to the right of left is (1 – w). The matrix M
is of the type known as circulant matrix. Such an n x n matrix is of the form
Co
C2
Cn-1
Сп-1
CO
Cn-2
C =
Сп-2 Сп-1
CO
Cn-3
C1
C2
C3
CO
with the normalized eigenvectors
1
1
e2mij/n
j = 1,...,n.
e2(n-1)xij/n
(i) Use this result to find the eigenvalues of the matrix C.
(ii) Use (i) to find the eigenvalues of the matrix M.
(iii) Use (ii) to find p(t) (t = 0,1, 2,...), where we expand the initial distribution
vector p(0) in terms of the eigenvectors
p(0) = arek with EP;(0):
= 1.
k=1
j=1
(iv) Assume that p(0) = (1 1
1)". Give the time evolution of p(0).
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