- 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
Solutions for Chapter 8.1: Stable Age Distribution in a Population; Markov ProcessesGet Full Solutions
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. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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 ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Skew-symmetric 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 A-I 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 = U-I.
Orthonormal columns (complex analog of Q).