- 3.5.1: The populations of U.S. metropolitan and nonmetropolitan areas in 2...
- 3.5.2: The populations of U.S. cities, suburbs, and nonmetro areas in 2012...
- 3.5.3: Consider a genetics model in which the offspring of guinea pigs are...
- 3.5.4: The statistics for rainfall for the month of December in Tel Aviv a...
- 3.5.5: A psychologist conducts an experiment in which 20 rats are placed a...
- 3.5.6: 40 rats are placed at random in a compartment having four rooms lab...
- 3.5.7: Two car rental companies A and B are competing for customers at cer...
- 3.5.8: A market research group has been studying the buying patterns for t...
- 3.5.9: Prove that 1 is an eigenvalue of every stochastic matrix. (Hint: Pr...
- 3.5.10: Consider a colony of beetles. Assume that the beetles live at most ...
- 3.5.11: Consider a colony of raccoons. Assume that the raccoons live at mos...
- 3.5.12: A laboratory is conducting an experiment to analyze growth of beetl...
Solutions for Chapter 3.5: Google, Demography, Weather Prediction, and Leslie Matrix Models
Full solutions for Linear Algebra with Applications | 8th Edition
Solutions for Chapter 3.5: Google, Demography, Weather Prediction, and Leslie Matrix ModelsGet Full Solutions
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.