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Questions and Answers of Linear Algebra

True or false: If B = S-1AS are similar matrices, then (a) ||B||∞ = ||A||∞ (b) ||B||2 = ||A||2 (c) p(B) = p(A)
Prove that the condition number of a nonsingular matrix is given by k(A) = ||A||2 ||A-1||2.
(i) Find an explicit formula for the 1 matrix norm ||A|||1.(ii) Compute the 1 matrix norm of the matrices in Exercise 10.3.1, and discuss convergence.
Prove directly from the axioms of Definition 3.12 that (10.40) defines a norm on the space of n × n matrices.
Let K > 0 be a positive definite matrix. Characterize the matrix norm induced by the inner product (x. y) = xT K y. Use Exercise 8.4.45.
LetCompute the matrix norm ||A||| using the following norms in R2: (a) The weighted ˆž norm ||v|| = max{2|v1|. 3|v2|| (b) The weighted 1 norm ||v|| = 2|v1| + 3 |v2|; (c) The weighted inner
The Frobenius norm of an n Ã— n matrix A is defined asProve that this defines a matrix norm by checking the three norm axioms plus the multiplicative inequality (10.33).
Let A be an n × n matrix with singular value vector σ = (σ1,... , σ1). Prove that (a) ||σ||∞ = ||A||2; (b) ||σ||2 = ||A||F, the Frobenius norm of Exercise 10.3.16. Remark: ||σ||2 (also
Explain why ||A|| = max|aij| defines a norm on the space of n x n matrices. Show by example that this is not a matrix norm, i.e., (10.33) is not necessarily valid.
Prove that the closed curve parametrized in (10.37) is an ellipse. What are its semi-axes?
Compute the Euclidean matrix norm of each matrix in Exercise 10.3.1. Have your convergence conclusions changed?In Exercise 10.3.1Compute the ˆž matrix norm of the following matrices. Which are
(a) Prove that the individual entries aij of a matrix A are bounded in absolute value by its ˆž matrix norm: |aij| (b) Prove that if the seriesconverges, then the matrix series converges to
(a) Use Exercise 10.3.20 to prove that the geometric matrix seriesconverges whenever p(A) (b) Prove that the sum is (I - A)-1. How do you know I - A is invertible?
For each of the following matrices,(i) Find all Gerschgorin disks;(ii) Plot the Gerschgorin domain in the complex plane;(iii) Compute the eigenvalues and confirm the truth of the Circle Theorem
(i) Explain why the eigenvalues of A must lie in its refined Gerschgorin domain D*A = DAT DA. (ii) Find the refined Gerschgorin domains for each of the matrices in Exercise 10.3.22 and confirm the
Let A be a square matrix. Prove that max {0, t} < p(A) < s, where s = max{s1,... , sn} is the maximal absolute row sum of A. as defined in (10.39), and t = min{|aij| - ri}. with ri given by (10.44).
(a) Suppose that every entry of the n Ã— n matrix A is bounded byProve that A is a convergent matrix. Use Exercise 10.3.25. (b) Produce a matrix of size n x n with one or more entries
Compute the spectral radii of the matrices in Exercise 10.3.1. Which are convergent? Compare your conclusions with those of Exercises 10.3.1 and 10.3.2.In Exercise 10.3.1Compute the ˆž
Prove that if K is symmetric, diagonally dominant. and each diagonal entry is positive, then K is positive definite.
Prove that if A is diagonally dominant and each diagonal entry is negative, then the zero equilibrium solution to the linear system of ordinary differential equations u = A u is asymptotically stable.
Let k be an integer and setCompute (a) ||Ak||ˆž (b) ||Ak||2 (c) p(Ak) (d) Explain why every Ak is a convergent matrix, even though their matrix norms can be arbitrarily large. (e) Why does
Find a matrix A such that (a) ||A2||∞ ≠ ||A||2∞ (b) ||A2||∞ ≠ ||A||22.
Find a convergent matrix that has dominant singular value σ1 > 1.
Determine if the following matrices are regular transition matrices. If so, find the associated probability eigenvector.(a)(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m)
A bug crawls along the edges of the pictured triangular lattice with six vertices. Upon arriving at a vertex, there is an equal probability of its choosing any edge to leave the vertex. Set up the
Answer Exercise 10.4.10 for the larger triangular lattice.In Exercise 10.4.10 A bug crawls along the edges of the pictured triangular lattice with six vertices. Upon arriving at a vertex, there is an
Suppose the bug of Exercise 10.4.10 crawls along the edges of the pictured square lattice. What can you say about its behavior?In Exercise 10.4.10 A bug crawls along the edges of the pictured
Prove that, for all 0 0, the probability eigenvector of the transition matrixIs
True or false: If T is a transition matrix, so is T-1.
True or false: Every probability eigenvector of a regular transition matrix has eigenvalue equal to 1.
Let T be a transition matrix. Prove that if u is a probability vector, so is v = T u.
(a) Prove that if T and 5 are transition matrices, so is their product T S. (b) Prove that if T is a transition matrix, so is Tk for any k > 0.
The population of an island is divided into city and country residents. Each year, 5% of the residents of the city move to the country and 15% of the residents of the country move to the city. In
A traveling salesman visits the three cities of Atlanta, Boston, and Chicago. The matrixdescribes the transition probabilities of his trips. Describe his travels in words, and calculate how often he
A business executive is managing three branches, labeled A, B and C, of a corporation. She never visits the same branch on consecutive days. If she visits branch A one day, she visits branch B the
Explain why the irregular Markov process with transition matrixdoes not reach a steady state. Use a population model as is Exercise 10.4.3 to interpret what is going on.
A genetic model describing inbreeding, in which mating takes place only between individuals of the same genotype, is given by the Markov process u(n+1) = T u(n), whereis the transition matrix
(a) Find the spectral radius of the matrix(b) Predict the long term behavior of the iterative system in as much detail as you can.
Consider the iterative system (10.52) with spectral radius p(T) < 1. Explain why it takes roughly - l/log10p(T) iterations to produce one further decimal digit of accuracy in the solution.
Consider the linear system A x = b, where(a) First, solve the equation directly by Gaussian Elimination. (b) Using the initial approximation x(0) = 0, carry out three iterations of the Jacobi
For the diagonally dominant systems in Exercise 10.5.3. starting with the initial guess x = y = z = 0, compute the solution to 3 decimal places using the Gauss-Seidel method. Check your answer by
Which of the systems in Exercise 10.5.3 lead to convergent Gauss-Seidel schemes? In each case, which converges faster, Jacobi or Gauss-Seidel? In Exercise 10.5.3 Which of the following systems have a
(a) Solve the positive definite linear systems in Exercise 10.5.6 using the Gauss-Seidel scheme to achieve 4 decimal place accuracy.(b) Compare the convergence rate with the Jacobi method.
Let A =(a) For what values of c is A diagonally dominant? (b) Use a computer to find the smallest positive value of c > 0 for which Jacobi iteration converges. (c) Find the smallest positive value of
Consider the linear system 2.4x - .8y + .8z = 1. - .6x + 3.6y - .6 z = 0. 15x + 14.4 y - 3.6z = 0. Show, by direct computation, that Jacobi iteration converges to the solution, but Gauss-Seidel does
Discuss convergence of Gauss-Seidel iteration for the system 5x + 7y + 6z + 5w = 23 7x + 10y + 8z + 7w = 32 6x + 8y + l0z + 9w = 33 5x + 7y + 9z + 10w = 31.
Answer Exercise 10.5.1 when(a)(b) (c)
How many arithmetic operations are needed to perform k steps of the Jacobi iteration? What about Gauss-Seidel? Under what conditions is Jacobi or Gauss-Seidel more efficient than Gaussian Elimination?
Consider the linear system Ax = e1 based on the 10 Ã— 10 pentadiagonal matrix(a) For what values of z are the Jacobi and Gauss- Seidel methods guaranteed to converge? (b) Set z = 4. How
The naive iterative method for solving Au = b is to rewrite it in fixed point form u = T u + c, where T = I - A and c = b.(a) What conditions on the eigenvalues of A ensure convergence of the naive
Consider the linear system Au = b, where(a) What is the solution? (b) Discuss the convergence of the Jacobi iteration method. (c) Discuss the convergence of the Gauss-Seidel iteration method. (d)
Consider the linear system 4x - y - z = 1 -x + 4y - w = 2 -x + 4z - w = 0 - y - z + 4w = 1. (a) Find the solution by using Gaussian Elimination and Back Substitution. (b) Using 0 as your initial
(a) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices when(b) Is A diagonally dominant? (c) Use (10.86) to fix the optimal value of the SOR parameter. Verify that the
Change the matrix in Exercise 10.5.26 toand answer the same questions. Does the SOR method with parameter given by (10.86) speed the iterations up? Why not? Can you find a value of the SOR parameter
Consider the linear system Au = e1 in which A is the 8 × 8 tridiagonal matrix with all 2's on the main diagonal and all - l's on the sub- and superdiagonal.(a) Use Exercise 8.2.47 to find the
In Exercise 10.5.18 you were asked to solve a system by Gauss-Seidel. How much faster can you design an SOR scheme to converge? Experiment with several values of the relaxation parameter co, and
Investigate the three basic iterative techniques-Jacobi, Gauss-Seidel. SOR-for solving the linear system K*u* = f* for the cubical circuit in Example 6.4.
The matrixA =arises in the finite difference (and finite element) discretization of the Poisson equation on a nine point square grid. Solve the linear system A u = e5 using (a) Gaussian
The generalization of Exercise 10.5.31 to an n Ã— n grid results in an n2 Ã— n2 matrix in block tridiagonal formin which K is the tridiagonal n Ã— n matrix with
How much can you speed up the convergence of the iterative solution to the pentadiagonal linear system in Exercise 10.5.22 when z = 4 by using SOR? Discuss.
For the matrix treated in Example 10.50, prove that (a) As ω increases from 1 to 8 - 4√3, the two eigenvalues move towards each other, with the larger one decreasing in magnitude; (b) if ω > 8 -
If u(k) is an approximation to the solution to A u = b, then the residual vector r(k) = b - Au(k) measures how accurately the approximation solves the system.(a) Show that the Jacobi iteration can be
Let K be a positive definite nxn matrix with eigenvalues λ1 > λ2 > ∙ ∙ ∙ > λn > 0. For what values of e does the iterative system u(k+1) = u(k) + εr(k), where r(k) = f - K u(k)
Solve the following linear systems by the conjugate gradient method, keeping track of the residual vectors and solution approximations as you iterate.(a)(b) (c) (d) (e)
Use the conjugate gradient method to solve the system in Exercise 10.5.31. How many iterations do you need to obtain the solution that is accurate to 2 decimal places? How does this compare to the
According to Example 3.33. the n × n Hilbert matrix Hn is positive definite, and hence we can apply the conjugate gradient method to solve the linear system Hnu = f. For the values n = 5. 10. 30,
For the diagonally dominant systems in Exercise 10.5.3, starting with the initial guess x = y = z = 0. Compute the solution to 2 decimal places using the Jacobi method. Check your answer by solving
Try applying the Conjugate Gradient Algorithm to the system -x + 2y + z = -2, y + 2z = 1. 3x + y - z = 1. Do you obtain the solution? Why?
True or false: If the residual vector satisfies ||r|| < .01, then u approximates the solution to within two decimal places.
How many arithmetic operations are needed to implement one iteration of the conjugate gradient method? How many iterations can you perform before the method becomes more work that direct Gaussian
In (10.90), find the value of tk that minimizes p(uk+1).
(a) Do any of the non-diagonally dominant systems in Exercise 10.5.3 lead to convergent Jacobi schemes? Check the spectral radius of the Jacobi matrix. (b) For the convergent systems in Exercise
The following linear systems have positive definite coefficient matrices. Use the Jacobi method starting with u(0) = 0 to find the solution to 4 decimal place accuracy.(a)(b) (c) (d) (e) (f)
Let A be the n × n tridiagonal matrix with all its diagonal entries equal to c and all 1 's on the sub- and super-diagonals.(a) For which values of c is A diagonally dominant?(b) For which values of
Prove that 0 ≠ u ∈ ker A if and only if u is a eigenvector of the Jacobi iteration matrix with eigenvalue 1. What does this imply about convergence?
Prove that if A is a nonsingular coefficient matrix, then one can always arrange that all its diagonal entries are nonzero by suitably permuting its rows.
Use the power method to find the dominant eigenvalue and associated eigenvector of the following matrices:(a)(b) (c) (d) (e) (f) (g) (h)
Suppose that Au(k) = 0 in the iterative procedure (10.101). What does this indicate?
Apply the Q R algorithm to the following symmetric matrices to find their eigenvalues and eigenvectors to 2 decimal places:(a)(b) (c) (d) (e) (f)
Show that applying the Q R algorithm to the matrixresults in a diagonal matrix with the eigenvalues on the diagonal, but not in decreasing order. Explain.
Apply the Q R algorithm to the following non- symmetric matrices to find their eigenvalues to 3 decimal places:(a)(b)(c)(d)(e)
The matrixhas a double eigenvalue of 1, and so our proof of convergence of the Q R algorithm doesn't apply. Does the Q R algorithm find its eigenvalues?
(a) Prove that if A is symmetric and tridiagonal, then all matrices Ak appearing in the Q R algorithm are also symmetric and tridiagonal. First prove symmetry. (b) Is the result true if A is not
Use Householder matrices to convert the following matrices into tridiagonal form:(a)(b) (c)
Find the eigenvalues, to 2 decimal places, of the matrices in Exercise 10.6.18 by applying the QR algorithm to the tridiagonal form.In Exercise 10.6.18Use Householder matrices to convert the
Let Tn be the tridiagonal matrix whose diagonal entries are all equal to 2 and whose sub- and superdiagonal entries all equal 1. Use the power method to find the dominant eigenvalue of Tn for n = 10,
Use the tridiagonal Q R method to find the singular values of the matrix
Use Householder matrices to convert the following matrices into upper Hessenberg form:(a)(b) (c)
Find the eigenvalues, to 2 decimal places, of the matrices in Exercise 10.6.21 by applying the QR algorithm to the upper Hessenberg form.In Exercise 10.6.21Use Householder matrices to convert the
Prove that the effect of the first Householder reflection is as given in (10.109).
What is the effect of tridiagonalization on the eigenvectors of the matrix?
(a) How many arithmetic operations (multiplications/divisions and additions/subtractions) are required to place a generic n x n symmetric matrix into tridiagonal form?(b) How many operations are
Write out a pseudocode program to tridiagonalize a matrix. The input should be an n × n matrix A and the output should be the Householder unit vectors u1.......un-1 and the tridiagonal matrix R.
Use the power method to find the largest singular value of the following matrices:(a)(b) (c) (d)
Prove that, for the iterative scheme (10.101), ||Au(k)|| → |λ1||. Assuming λ1 is real, explain how to deduce its sign.
The Inverse Power Method. Let A be a nonsingular matrix. (a) Show that the eigenvalues of A-1 are the reciprocals 1/λ of the eigenvalues of A. How are the eigenvectors related? (b) Show how to use
Apply the inverse power method of Exercise 10.6.7 to the find the smallest eigenvalue of the matrices in Exercise 10.6.1.In Exercise 10.6.1Use the power method to find the dominant eigenvalue and
The Shifted Inverse Power Method. Suppose that u is not an eigenvalue of A. (a) Show that the iterative scheme u(k+1) = (A - μ I)-1 u(k) converges to the eigenvector of A corresponding to the
Apply the shifted inverse power method of Exercise 10.6.7 to the find the eigenvalue closest to Î¼ = .5 of the matrices in Exercise 10.6.1.In Exercise 10.6.1Use the power method to find
(i) Explain how to use the deflation method of Exercise 8.2.52 to find the subdominant eigenvalue of a nonsingular matrix A.(ii) Apply your method to the matrices in Exercise 10.6.1.In Exercise

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