 5.5.1: Tile row un~; of the tr:tnsition matrix of a Markov chain are :111 l
 5.5.2: If the transition matrix of a Markov chain contains zero emries. th...
 5.5.3: . If A is the tr:msition matrix of a Markov chain and p is any prob...
 5.5.4: f A is the tr:msition 1mttrix of a Markov chain and p is any probab...
 5.5.5: f A is the transition matrix of a regular Markov chain, then as m a...
 5.5.6: Every regular tranSitiOn matrix has I as an eigenvalue
 5.5.7: very regular transition matrix has a unique probability vector that...
 5.5.8: The general solution of y' = ky
 5.5.9: If p  I AP is a diagon:1l matrix D. then the change of variable z ...
 5.5.10: A differential equation a.ly"' + a2y" + cqy' + ao.v = 0, where lll,...
 5.5.11: If A =PDP '. where P i~ an inveltible matrix and D is a diagonal m...
 5.5.12: n a Fibonacci sequence. each term after the fir.t two is the sum of...
 5.5.13: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.14: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.15: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.16: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.17: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.18: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.19: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.20: In EurdSI!S 13 20, dt'termill<' 1\'llether ettcll tmnsitionmatri.x...
 5.5.21: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.22: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.23: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.24: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.25: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.26: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.27: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.28: In Exercises 21 28. a IY!8ular transition matrix is given. Determi...
 5.5.29: When Ali>on goe' to her fa\'Orite ice cream store. she bu)> etther ...
 5.5.30: Suppose that the probability that the child of a collegeeducmed P ...
 5.5.31: A supertn:ll"ket >ells three brands of baking powder. Of those who ...
 5.5.32: Suppose that a panicular region with a constant population is divid...
 5.5.33: Prove that lhe sum of the e ntries of each column of the matrix A i...
 5.5.34: Verify that the matrix A in the subsection on Googlc searches satis...
 5.5.35: In [5]. Gabriel and Neumann found that a Markov chain could be used...
 5.5.36: A company leases rental cars at three Chicago offices (located at M...
 5.5.37: Suppose that the tnmsition mmrix of a Markov chain is [ .90 . I A =...
 5.5.38: Gi\e an example of a 3 x 3 regular tr:tn\ition matrix A such that A...
 5.5.39: Let A be an 11 x 11 Mocha.,tic matrix. and let u be the vector in '...
 5.5.40: Usc ideas from Exercise 37 to con,truct two regular 3 x 3 'tocha,ti...
 5.5.41: Prove that if A is a stochastic matrix and p is a probability vecto...
 5.5.42: Let A be the 2 x 2 stocha.>tic matrix [ 1 (I I b] b . (a) Detenni...
 5.5.43: Let A be an 11 x 11 ~tocha,tic matrix. (a) Let v be any vector m R"...
 5.5.44: Prove that if A and 8 arc >tochastic matrices. then AB is a stochas...
 5.5.45: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.46: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.47: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.48: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.49: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.50: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.51: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.52: /11 Exen:ises 45 52. ji11d rite geneml .wlurio11 of each sysrem of...
 5.5.53: In Exercises 5360. find the particular solution of each system of ...
 5.5.54: In Exercises 5360. find the particular solution of each system of ...
 5.5.55: In Exercises 5360. find the particular solution of each system of ...
 5.5.56: In Exercises 5360. find the particular solution of each system of ...
 5.5.57: In Exercises 5360. find the particular solution of each system of ...
 5.5.58: In Exercises 5360. find the particular solution of each system of ...
 5.5.59: In Exercises 5360. find the particular solution of each system of ...
 5.5.60: In Exercises 5360. find the particular solution of each system of ...
 5.5.61: Convert the secondorder differential equation y"  2y'  3y = 0 in...
 5.5.62: Convert the thirdorder differential equation y"'  2y"  8y' = 0 i...
 5.5.63: Convert the thirdorder differential equation y"'  2y"  y' + 2y =...
 5.5.64: Find the general solution of the differential equation that describ...
 5.5.65: Find the general solution of the differential equation that describ...
 5.5.66: Let z be ;m 11 x I column vector of differentiable functions. and l...
 5.5.67: Let Yt denote the number of rabbits in a certain area at time t and...
 5.5.68: Show that the characteristic polynomial of [_~ is 13 at2 bt c.
 5.5.69: Show that if )q, ),2 . and ),3 are distinct roots o r the polynomia...
 5.5.70: In Erercises 7078, use either of the two methods developed in this...
 5.5.71: In Erercises 7078, use either of the two methods developed in this...
 5.5.72: In Erercises 7078, use either of the two methods developed in this...
 5.5.73: In Erercises 7078, use either of the two methods developed in this...
 5.5.74: In Erercises 7078, use either of the two methods developed in this...
 5.5.75: In Erercises 7078, use either of the two methods developed in this...
 5.5.76: In Erercises 7078, use either of the two methods developed in this...
 5.5.77: In Erercises 7078, use either of the two methods developed in this...
 5.5.78: In Erercises 7078, use either of the two methods developed in this...
 5.5.79: Suppose 1ha1 we have a large number of blocks. The blocks are of fi...
 5.5.80: Suppo'e 1hat a bank pay' inlere>l on >:JVing' of 8% compounded annu...
 5.5.81: Wrile lhe thirdorder difference equ:lliun r. = 4r._,  2r. 2 +Sr. ...
 5.5.82: Justify the matrix fonn of equation Sl ghen in this >cction: s. = A...
 5.5.83: Consider a klhorder difference equation in the fonn of equation l ...
 5.5.84: Prove 1h:u a scalar A is an eigenvalue of lhe malrix A in Exerci~e ...
 5.5.85: ft1 Exercises 85 87. we examine rlre nonhomogeneous firM order dif...
 5.5.86: ft1 Exercises 85 87. we examine rlre nonhomogeneous firM order dif...
 5.5.87: ft1 Exercises 85 87. we examine rlre nonhomogeneous firM order dif...
 5.5.88: An invcs1or opened a savings accoum on March I with an inillal depo...
 5.5.89: Sohe the system of differential equations y; = 3.2)'t + 4. h2 + 7.7...
 5.5.90: In (3], Bourne examined the changes in land use in Toronto. Canada....
 5.5.91: A search cngmc consoders only 10 Web pages. which arc linked in the...
Solutions for Chapter 5.5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 5.5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION
Get Full SolutionsChapter 5.5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION includes 91 full stepbystep solutions. Since 91 problems in chapter 5.5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION have been answered, more than 25837 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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 v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.