 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: (Ps)"cJlOlogy) A bclilvioral psychologist places a rnt each day in ...
 8.1.11: (S/Jci/Jlogy) A study ha~ determined thaI the occupation of a boy. ...
 8.1.12: (Gel1etics) Consider a plant that can have red flowers (R). pink fl...
 8.1.13: (Mass Trallsit) A new mass transit system has just gone into operat...
Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 8.1: Stable Age Distribution in a Population; Markov Processes includes 13 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 8.1: Stable Age Distribution in a Population; Markov Processes have been answered, more than 13627 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).