Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Table of Contents
1.1
Vectors and Linear Combinations
1.2
Lengths and Dot Products
1.3
Matrices
2.1
Solving Linear Equations
2.2
The Idea of Elimination
2.3
Elimination Using Matrices
2.4
Rules for Matrix Operations
2.5
Inverse Matrices
2.6
Elimination = Factorization: A = L U
2.7
Transposes and Permutations
3.1
Spaces of Vectors
3.2
The Nullspace of A: Solving Ax = 0
3.3
The Rank and the Row Reduced Form
3.4
The Complete Solution to Ax = b
3.5
Independence, Basis and Dimension
3.6
Dimensions of the Four Subspaces
4.1
Orthogonality of the Four Subspaces
4.2
Projections
4.3
Least Squares Approximations
4.4
Orthogonal Bases and Gram-Schmidt
5.1
The Properties of Determinants
5.2
Permutations and Cofactors
5.3
Cramer's Rule, Inverses, and Volumes
6.1
Introduction to Eigenvalues
6.2
Diagonalizing a Matrix
6.3
Applications to Differential Equations
6.4
Symmetric Matrices
6.5
Positive Definite Matrices
6.6
Similar Matrices
6.7
Singular Value Decomposition (SVD)
7.1
The Idea of a Linear Transformation
7.2
The Matrix of a Linear Transformation
7.3
Diagonalization and the Pseudoinverse
8.1
Matrices in Engineering
8.2
Graphs and Networks
8.3
Markov Matrices, Population, and Economics
8.4
Linear Programming
8.5
Fourier Series: Linear Algebra for Functions
8.6
Linear Algebra for Statistics and Probability
8.7
Computer Graphics
9.1
Gaussian Elimination in Practice
9.2
Norms and Condition Numbers
9.3
Iterative Methods and Preconditioners
10.1
Complex Numbers
10.2
Hermitian and Unitary Matrices
10.3
The Fast Fourier Transform
Textbook Solutions for Introduction to Linear Algebra
Chapter 8.3 Problem 18
Question
For the Leslie matrix show that det(A -AI) = gives FIA 2 + F2P1A + F3Pl P2 = A 3 . The right side A 3 is larger as A ----+ 00. The left side is larger at A = 1 if Fl + F2 PI + F3 PI P2 > 1. In that case the two sides are equal at an eigenvalue Amax > 1: growth.
Solution
The first step in solving 8.3 problem number 18 trying to solve the problem we have to refer to the textbook question: For the Leslie matrix show that det(A -AI) = gives FIA 2 + F2P1A + F3Pl P2 = A 3 . The right side A 3 is larger as A ----+ 00. The left side is larger at A = 1 if Fl + F2 PI + F3 PI P2 > 1. In that case the two sides are equal at an eigenvalue Amax > 1: growth.
From the textbook chapter Markov Matrices, Population, and Economics you will find a few key concepts needed to solve this.
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full solution
Title
Introduction to Linear Algebra 4
Author
Gilbert Strang
ISBN
9780980232714
For the Leslie matrix show that det(A -AI) = gives FIA 2 + F2P1A + F3Pl P2 = A 3 . The
Chapter 8.3 textbook questions
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Chapter 8: Problem 1 Introduction to Linear Algebra 4
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Chapter 8: Problem 2 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 3 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 4 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 5 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 6 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 7 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 8 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 9 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 10 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 11 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 12 Introduction to Linear Algebra 4
Questions 1-12 are about Markov matrices and their eigenvalues and powers.
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Chapter 8: Problem 13 Introduction to Linear Algebra 4
Questions 13-15 are about linear algebra in economics.
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Chapter 8: Problem 14 Introduction to Linear Algebra 4
Questions 13-15 are about linear algebra in economics.
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Chapter 8: Problem 15 Introduction to Linear Algebra 4
Questions 13-15 are about linear algebra in economics.
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Chapter 8: Problem 16 Introduction to Linear Algebra 4
(Markov again) This matrix has zero determinant. What are its eigenvalues? [ .4 .2 .3] A = .2 .4 .3 . .4 .4 .4 Find the limits of Akuo starting from Uo = (1,0,0) and then Uo = (100,0,0).
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Chapter 8: Problem 17 Introduction to Linear Algebra 4
If A is a Markov matrix, does I + A + A2 + ... add up to (/ - A)-I?
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Chapter 8: Problem 18 Introduction to Linear Algebra 4
For the Leslie matrix show that det(A -AI) = gives FIA 2 + F2P1A + F3Pl P2 = A 3 . The right side A 3 is larger as A ----+ 00. The left side is larger at A = 1 if Fl + F2 PI + F3 PI P2 > 1. In that case the two sides are equal at an eigenvalue Amax > 1: growth.
Read more -
Chapter 8: Problem 19 Introduction to Linear Algebra 4
Sensitivity of eigenvalues: A matrix change I::!. A produces eigenvalue changes I::!. A. The formula for those changes 1::!.)1.1,".' I::!.An is diag(S-1 I::!. A S). Challenge: Start from AS = SA. The eigenvectors and eigenvalues change by I::!.S and I::!. A : (A+I::!.A)(S+~S) = (S+I::!.S)(A+I::!.A) becomes A(I::!.S)+(I::!.A)S = S(I::!.A)+(I::!.S)A. Small terms (I::!.A)(~S) and (ilS)(I::!.A) are ignored. Multiply the last equation by S-I. From the inner terms, the diagonal part of S-I(I::!.A)S gives ilA as we want. Why do the outer terms S-1 A I::!.S and S-1 I::!.S A cancel on the diagonal? Explain S-1 A = AS-1 and then diag(A S-1 I::!.S) = diag(S-1 I::!.S A).
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Chapter 8: Problem 20 Introduction to Linear Algebra 4
Suppose B > A > 0, meaning that each bij > aij > 0. How does the PerronFrobenius discussion show that Amax(B) > AmaxCA)?
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