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Matrix : Convergence, Pseudoinverse and Single Value Decomposition

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.

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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 ...

Solution Summary

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

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