 3.5.1: In Exercises 14, determine if A is a stochastic matrix.A =0.2 0.60....
 3.5.2: In Exercises 14, determine if A is a stochastic matrix.A = 1.5 0.15...
 3.5.3: In Exercises 14, determine if A is a stochastic matrix.A =15 1 025 ...
 3.5.4: In Exercises 14, determine if A is a stochastic matrix.A =314 38 12...
 3.5.5: In Exercises 58, fill in the missing values to make A a stochastic ...
 3.5.6: In Exercises 58, fill in the missing values to make A a stochastic ...
 3.5.7: In Exercises 58, fill in the missing values to make A a stochastic ...
 3.5.8: In Exercises 58, fill in the missing values to make A a stochastic ...
 3.5.9: In Exercises 912, if possible, fill in the missing values to make A...
 3.5.10: In Exercises 912, if possible, fill in the missing values to make A...
 3.5.11: In Exercises 912, if possible, fill in the missing values to make A...
 3.5.12: In Exercises 912, if possible, fill in the missing values to make A...
 3.5.13: In Exercises 1316, find the state vector x3 for the given stochasti...
 3.5.14: In Exercises 1316, find the state vector x3 for the given stochasti...
 3.5.15: In Exercises 1316, find the state vector x3 for the given stochasti...
 3.5.16: In Exercises 1316, find the state vector x3 for the given stochasti...
 3.5.17: In Exercises 1720, find all steadystate vectors for the givenstoch...
 3.5.18: In Exercises 1720, find all steadystate vectors for the givenstoch...
 3.5.19: In Exercises 1720, find all steadystate vectors for the givenstoch...
 3.5.20: In Exercises 1720, find all steadystate vectors for the givenstoch...
 3.5.21: In Exercises 2124, determine if the given stochastic matrix is regu...
 3.5.22: In Exercises 2124, determine if the given stochastic matrix is regu...
 3.5.23: In Exercises 2124, determine if the given stochastic matrix is regu...
 3.5.24: In Exercises 2124, determine if the given stochastic matrix is regu...
 3.5.25: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.26: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.27: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.28: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.29: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.30: FIND AN EXAMPLE For Exercises 2530, find an example thatmeets the g...
 3.5.31: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.32: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.33: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.34: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.35: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.36: TRUE OR FALSE For Exercises 3136, determine if the statementis true...
 3.5.37: . Prove Theorem 3.27(a): Show that each state vector is aprobabilit...
 3.5.38: Prove Theorem 3.27(b): Show that the product of two stochasticmatri...
 3.5.39: . Prove Theorem 3.27(c): Show that if A is a stochasticmatrix, then...
 3.5.40: Suppose that A is a regular stochastic matrix. Show that A2 isalso ...
 3.5.41: Let A =a bc dbe a doubly stochastic matrix. Prove thata = d and b = c.
 3.5.42: If A =0 11 0, prove that x0 =0.50.5is the only initial statevector ...
 3.5.43: Let A be a regular stochastic matrix, and suppose that k isthe smal...
 3.5.44: Let A be an upper or lower triangular stochastic matrix. Showthat A...
 3.5.45: Suppose that A = 0(1 ) 1, where 0 << 1.(a) Explain why A is a stoch...
 3.5.46: C In an office complex of 1000 employees, on any given daysome are ...
 3.5.47: C The star quarterback of a university football team hasdecided to ...
 3.5.48: C It has been claimed that the best predictor of todaysweather is y...
 3.5.49: C Consumers in Shelbyville have a choice of one of twofast food res...
 3.5.50: C An assembly line turns out two types of pastries, ChocolateZots a...
 3.5.51: C A mediumsize town has three public library branches,designated A...
 3.5.52: . C Let A =101/3011/3001/3.Numerically verify that each initial sta...
 3.5.53: C For Exercises 5354, determine to six decimal places the steadysta...
 3.5.54: C For Exercises 5354, determine to six decimal places the steadysta...
 3.5.55: C For Exercises 5556, use computational experimentation tofind two ...
 3.5.56: C For Exercises 5556, use computational experimentation tofind two ...
Solutions for Chapter 3.5: Markov Chains
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 3.5: Markov Chains
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. Since 56 problems in chapter 3.5: Markov Chains have been answered, more than 15043 students have viewed full stepbystep solutions from this chapter. Chapter 3.5: Markov Chains includes 56 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.