 8.1.1: Consider a living organism that can live to a maximum age of 2 year...
 8.1.2: Consider a living organism that can live 10 a maximum age of 2 year...
 8.1.3: Which of the following can be transition matrices of a Markov proce...
 8.1.4: Which of the following are probability vectors? la) m (,) m Ih) m (...
 8.1.5: Consider the trunsition matrix T = [0.7 0.4]. 0.3 0.6 (a) If ~O! = ...
 8.1.6: Consider the trnnsition matrix (a) If 0.2 0.3 0.5 0.0] 0.3 . 0.7 co...
 8.1.7: Which of the following tnmsilion matrices are regular? la) [~ : ] [...
 8.1.8: Show that each of the following trnnsition matrices reaches a state...
 8.1.9: Find the steadys ~lle vector of each of the following reg ular matr...
 8.1.10: A bclilvioral psychologist places a rnt each day in a cage wilh two...
 8.1.11: A study ha~ determined thaI the occupation of a boy. as an adult de...
 8.1.12: Consider a plant that can have red flowers (R). pink flowers (P). o...
 8.1.13: A new mass transit system has just gone into operation. The transit...
Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: Stable Age Distribution in a Population; Markov Processes includes 13 full stepbystep solutions. Since 13 problems in chapter 8.1: Stable Age Distribution in a Population; Markov Processes have been answered, more than 10104 students have viewed full stepbystep solutions from this chapter.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.