Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Problems 1-12 are about iterative methods for Ax = b.

Introduction to Linear Algebra 4
Divide equation (11) by A 1 and explain why I A2 j A 1 I controls the convergence of the power method. Construct a matrix A for which this method does not converge

Introduction to Linear Algebra 4
The Markov matrix A = [:~ :~] has A = 1 and .6, and the power method Uk = A k Uo converges to [:i~]. Find the eigenvectors of A-I. What does the inverse power method U-k = A-k Uo converge to (after you multiply by .6k )?

Introduction to Linear Algebra 4
The tridiagonal matrix of size n - 1 with diagonals -1, 2, -1 has eigenvalues Aj = 2 - 2cos(jrrjn). Why are the smallest eigenvalues approximately (jrrjn)2? The inverse power method converges at the speed Al j A2 ~ 1 j 4.

Introduction to Linear Algebra 4
For A = [-i -1] apply the power method Uk+l = AUk three times starting with Uo = [A]. What eigenvector is the power method converging to?

Introduction to Linear Algebra 4
In Problem 11 apply the inverse power method uk+ 1 = A-I Uk three times with the same uo. What eigenvector are the Uk'S approaching?

Introduction to Linear Algebra 4
In the QR method for eigenvalues, show that the 2,1 entry drops from sin e in A = QR to -sin3 e in RQ. (Compute Rand RQ.) This "cubic convergence" makes the method a success: Sine] = QR = [co.se -Sine] [1 ?] o sm e cos eo? .

Introduction to Linear Algebra 4
If A is an orthogonal matrix, its Q R factorization has Q = and R = __ Therefore RQ = . These are among the rare examples when the QR method goes nowhere.

Introduction to Linear Algebra 4
The shifted QR method factors A - cI into QR. Show that the next matrix Al - R Q + c I equals Q -1 A Q. Therefore A 1 has the eigenvalues as A (but is closer to triangular).

Introduction to Linear Algebra 4
When A = AT, the "Lanczos method" finds a's and b's and orthonormal q's so that Aqj = bj - 1q j-l +a jqj +b jq j+l (with qo = 0). Multiply by q} to find a formula for a j. The equation says that A Q = Q T where T is a tridiagonal matrix.

Introduction to Linear Algebra 4
The equation in Problem 21 develops from this loop with bo = 1 and r 0 = any q 1 : Write a code and test it on the -1, 2, -1 matrix A. QT Q should be I.

Introduction to Linear Algebra 4
Suppose A is tridiagonal and symmetric in the QR method. From Al _ Q-I AQ show that Al is symmetric. Write Al = RAR-I to show that Al is also tridiagonal. (If the lower part of A I is proved tridiagonal then by symmetry the upper part is too.) Symmetric tridiagonal matrices are the best way to start

Introduction to Linear Algebra 4
Questions 24-26 are about quick ways to estimate the location of the eigenvalues.

Introduction to Linear Algebra 4
Questions 24-26 are about quick ways to estimate the location of the eigenvalues.

Introduction to Linear Algebra 4
Questions 24-26 are about quick ways to estimate the location of the eigenvalues.

Introduction to Linear Algebra 4
Problems 27-30 present two fundamental iterations. Each step involves Aq or Ad. The key point for large matrices is that matrix-vector multiplication is much faster than matrix-matrix multiplication. A crucial construction starts with a vector b. Repeated mUltiplication will produce Ab, A 2 b, ... bu

Introduction to Linear Algebra 4
Problems 27-30 present two fundamental iterations. Each step involves Aq or Ad. The key point for large matrices is that matrix-vector multiplication is much faster than matrix-matrix multiplication. A crucial construction starts with a vector b. Repeated mUltiplication will produce Ab, A 2 b, ... bu

Introduction to Linear Algebra 4
Problems 27-30 present two fundamental iterations. Each step involves Aq or Ad. The key point for large matrices is that matrix-vector multiplication is much faster than matrix-matrix multiplication. A crucial construction starts with a vector b. Repeated mUltiplication will produce Ab, A 2 b, ... bu

Introduction to Linear Algebra 4
Problems 27-30 present two fundamental iterations. Each step involves Aq or Ad. The key point for large matrices is that matrix-vector multiplication is much faster than matrix-matrix multiplication. A crucial construction starts with a vector b. Repeated mUltiplication will produce Ab, A 2 b, ... bu