Purchase Solution

Matrix : Convergence, Pseudoinverse and Single Value Decomposition

Not what you're looking for?

Ask Custom Question

Only problems #3 &4-a,(without using any software).

3 .
Ax = b
we consider the iterative scheme
....
where the matrix Q is nonsingular.
(a) If ... for some subordinate matrix norm, show that the sequence produced by the above scheme converges to the solution of the system for any initial vector x(0).

4. Given singular matrix
....
(a) Find its singular-value decomposition.
(b) Find its pseudoinverse.

Attachments
Purchase this Solution
Solution Summary

Matrix problems involving Convergence, Pseudoinverse and Single Value Decomposition are solved. The solution is detailed and well presented.

Solution Preview

Hello and thank you for posting your question to Brainmass!
The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

The proofs for the theorems were taken from Atkinson

The numerical work was verified by MAPLE

Definition:

Let A be an arbitrary square matrix of order n, the spectrum of A is the set of all the eigenvalues, and it is denoted as (A). The spectral radius is the maximum size of these eigenvalues and it is denoted by:

Theorem 1:
Let A be an arbitrary square matrix of order n. then for every matrix norm we have:

.

Proof:

Let  be the largest eigenvalue of A (in absolute terms) and let v be the eigenvector associated with this eigenvalue with .
Then:

Theorem 2:

Let A be an arbitrary square matrix of order n. Then Am converges to the zero matrix as m→∞ if and only if ...

Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Perform Singular Value Decomposition

    504435 Perform Singular Value Decomposition I need singular value decomposition of the matrix A=[ 1 4 2 -2 -1 3

  • Least squares method

    Depending on the type of fit and initial parameters chosen, the nonlinear fit may have good or poor convergence properties.

  • Matrices

    Verify the special theorem for the Hermitian matrix see attached This provides examples of working with matrices, including eigenvalues and singular values, singular decomposition, and Hermitian matrix.

  • Spectral decomposition

    Consider the following (3x3) matrix A: Find the eigenvalues and eigenvectors of A and hence its spectral decomposition. Hence find .

  • Matrix Decomposition : LU Factorizations, Gram-Schmidt and Gershgorin's Theorem

    Matrix Decomposition, LU Factorizations, Gram-Schmidt and Gershgorin's Theorem are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

  • Hilbert matrices

    The solution shows how to work with Hilbert matrices to solve a system of equations using Gaussian elimination and to find the Doolittle LU decomposition with scaled partial pivoting.

  • Convergence = unique limit

    This way, the columns span all of space, and the planes formed by the rows intersect at a single point. Example: b) Temporary Failure. The matrix may need a row exchange to allow for a pivot to be found.

  • Spectral Decomposition and Conic Section

    = 0 ** Please see the attachment for the complete solution ** In this solution, we compute the spectral distribution of a matrix and compute and graph a conic section given by a quadratic equation in two variables.

  • Matlab to solve numerical linear algebra LU-decomposition

    Use LU decomposition to solve Ax = b Please see the attached file. This provides a Matlab file to solve given Gaussian elimination and LU decomposition problems.

View More Related Solutions