Problems 1-3 compute the SVD of a square singular matrix A.
Read more
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 6.7 Problem 16
Question
Run a random walk x (2), ... , x (n) starting from web site x (1) = 1. Count the visits to each site. At each step the code chooses the next website x (k) with probabilities given by column x(k - 1) of A. At the end, p gives the fraction of time at each site from a histogram: count visits. The ran kings are based on p. Please compare p to the steady state eigenvector of the Markov matrix A: A = [0 .1 .2 .7; .05 0 .15 .8; .15 .25 0 .6; .1 .3 .6 0]' n = 100; x = zeros(l,n); x(l) = 1; for k = 2 : n x(k) = min(find(rand
Solution
The first step in solving 6.7 problem number 16 trying to solve the problem we have to refer to the textbook question: Run a random walk x (2), ... , x (n) starting from web site x (1) = 1. Count the visits to each site. At each step the code chooses the next website x (k) with probabilities given by column x(k - 1) of A. At the end, p gives the fraction of time at each site from a histogram: count visits. The ran kings are based on p. Please compare p to the steady state eigenvector of the Markov matrix A: A = [0 .1 .2 .7; .05 0 .15 .8; .15 .25 0 .6; .1 .3 .6 0]' n = 100; x = zeros(l,n); x(l) = 1; for k = 2 : n x(k) = min(find(rand
From the textbook chapter Singular Value Decomposition (SVD) you will find a few key concepts needed to solve this.
Visible to paid subscribers only
Step 3 of 7)Visible to paid subscribers only
Subscribe to view the
full solution
full solution
Title
Introduction to Linear Algebra 4
Author
Gilbert Strang
ISBN
9780980232714
Run a random walk x (2), ... , x (n) starting from web site x (1) = 1. Count the visits
Chapter 6.7 textbook questions
-
Chapter 6: Problem 1 Introduction to Linear Algebra 4
-
Chapter 6: Problem 2 Introduction to Linear Algebra 4
Problems 1-3 compute the SVD of a square singular matrix A.
Read more -
Chapter 6: Problem 3 Introduction to Linear Algebra 4
Problems 1-3 compute the SVD of a square singular matrix A.
Read more -
Chapter 6: Problem 4 Introduction to Linear Algebra 4
Problems 4-7 ask for the SVD of matrices of rank 2.
Read more -
Chapter 6: Problem 5 Introduction to Linear Algebra 4
Problems 4-7 ask for the SVD of matrices of rank 2.
Read more -
Chapter 6: Problem 6 Introduction to Linear Algebra 4
Problems 4-7 ask for the SVD of matrices of rank 2.
Read more -
Chapter 6: Problem 7 Introduction to Linear Algebra 4
Problems 4-7 ask for the SVD of matrices of rank 2.
Read more -
Chapter 6: Problem 8 Introduction to Linear Algebra 4
A square invertible matrix has A-I = V :E-1 UT This says that the singular values of A-I are l/(J(A). Show that (Jmax(A- 1) (Jmax(A) > 1.
Read more -
Chapter 6: Problem 9 Introduction to Linear Algebra 4
Suppose Ul, ... ,Un and Vb .. . ,Vn are orthononnal bases for Rn. ,Construct the matrix A that transfonns each V j into U j to give AVI = Ul, ... ,Avn = Un.
Read more -
Chapter 6: Problem 10 Introduction to Linear Algebra 4
Construct the matrix with rank one that has Av 12u for v = !(1, 1, 1, 1) and u = 1(2,2,1). Its only singular value is (Jl = __
Read more -
Chapter 6: Problem 11 Introduction to Linear Algebra 4
Suppose A has orthogonal columns WI, W2, .. . ,Wn of lengths (Jl, (J2, ... ,(In. What are U, :E, and V in the SVD?
Read more -
Chapter 6: Problem 12 Introduction to Linear Algebra 4
Suppose A is a 2 by 2 symmetric matrix with unit eigenvectors UI and U2. If its eigenvalues are Al = 3 and A2 = -2, what are the matrices U, :E, VT in its SVD?
Read more -
Chapter 6: Problem 13 Introduction to Linear Algebra 4
If A = QR with an orthogonal matrix Q, the SVD of A is almost the same as the SVD of R. Which of the three matrices U,:E, V is changed because of Q?
Read more -
Chapter 6: Problem 14 Introduction to Linear Algebra 4
Suppose A is invertible (with (Jl > (J2 > 0). Change A by as small a matrix as possible to produce a singular matrix Ao. Hint: U and V do not change: From A = [UI U2] [(Jl (J2] [VI V2 r find the nearest Ao
Read more -
Chapter 6: Problem 15 Introduction to Linear Algebra 4
Why doesn't the SVD for A + I just use 2: + I?
Read more -
Chapter 6: Problem 16 Introduction to Linear Algebra 4
Run a random walk x (2), ... , x (n) starting from web site x (1) = 1. Count the visits to each site. At each step the code chooses the next website x (k) with probabilities given by column x(k - 1) of A. At the end, p gives the fraction of time at each site from a histogram: count visits. The ran kings are based on p. Please compare p to the steady state eigenvector of the Markov matrix A: A = [0 .1 .2 .7; .05 0 .15 .8; .15 .25 0 .6; .1 .3 .6 0]' n = 100; x = zeros(l,n); x(l) = 1; for k = 2 : n x(k) = min(find(rand
Read more -
Chapter 6: Problem 17 Introduction to Linear Algebra 4
The 1, -1 first difference matrix A has AT A = second difference matrix. The singular vectors of A are sine vectors v and cosine vectors u. Then Av = (JU is the discrete form of d/dx(sincx) = c(coscx). This is the best SVD I have seen. 1 0 0 SYDor A A= -1 I 0 o -1 1 o 0-1 ATA= -1 2-1 [ 2 -1 0] Orthogonal sine matrix 1 [Sinn/4 V = - sin2n/4 ,Ji sin 3n / 4 o -1 2 sin 2n / 4 sin 3n I 4] sin4n/4 sin6nl4 sin 6n / 4 sin 9n / 4 (a) Put numbers in V: The unit eigenvectors of AT A are singular vectors of A. Show that the columns of V have AT Av = AV with A = 2 - ,Ji, 2, 2 + ,Ji. (b) Multiply AV and verify that its columns are orthogonal. They are (JlUl and (J2U2 and (J3U3. The first columns of the cosine matrix U are Ul, U2, U3. (c) Since A is 4 by 3, we need a fourth orthogonal vector U4. It comes from the nullspace of AT. What is U4? The cosine vectors in U are eigenvectors of AAT. The fourth cosine is (1, 1, 1, 1)/2. 1 -1 0 0 cos n 18 cos 2n /8 cos 3 n / 8 AAT = -1 2 -IOU = _1_ cos3n/8 cos6n/8 cos9n/8 o -1 2 -1 ,Ji cos5n/8 cos lOnl8 cos 15nl8 o 0 -1 1 cos7n/8 cos 14n/8 cos21n/8 Those angles n /8, 3n 18, 5n /8, 7n /8 fit 4 points with spacing n / 4 between 0 and n. The sine transform has three points n / 4, 2n / 4, 3n / 4. The full cosine transform includes U4 from the "zero frequency" or direct current eigenvector (1, 1, 1, 1). The 8 by 8 cosine transform in 2D is the workhorse of jpeg compression. Linear algebra (circulant, Toeplitz, orthogonal matrices) is at the heart of signal processing.
Read more