 8.3.1: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.2: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.3: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.4: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.5: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.6: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.7: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.8: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.9: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.10: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.11: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.12: Questions 112 are about Markov matrices and their eigenvalues and ...
 8.3.13: Questions 1315 are about linear algebra in economics.
 8.3.14: Questions 1315 are about linear algebra in economics.
 8.3.15: Questions 1315 are about linear algebra in economics.
 8.3.16: (Markov again) This matrix has zero determinant. What are its eigen...
 8.3.17: If A is a Markov matrix, does I + A + A2 + ... add up to (/  A)I?
 8.3.18: For the Leslie matrix show that det(A AI) = gives FIA 2 + F2P1A + ...
 8.3.19: Sensitivity of eigenvalues: A matrix change I::!. A produces eigenv...
 8.3.20: Suppose B > A > 0, meaning that each bij > aij > 0. How does the Pe...
Solutions for Chapter 8.3: Markov Matrices, Population, and Economics
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 8.3: Markov Matrices, Population, and Economics
Get Full SolutionsSince 20 problems in chapter 8.3: Markov Matrices, Population, and Economics have been answered, more than 11500 students have viewed full stepbystep solutions from this chapter. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Chapter 8.3: Markov Matrices, Population, and Economics includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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