3. Let A be an n x n symmetric matrix that is singular. Consider the iteration M

Question

3. Let A be an n x n symmetric matrix that is singular. Consider the iteration Mx^(k+1) = Nx^(k) + b. (a) Show that at least one eigenvalue of B = M^-1N equals 1. (b) Describe how you could modify a convergent algorithm to obtain a solution to a singular system. Hints: For general b, Ax = b does not have a solution; in such cases, your algorithm should instead solve Ax = bar b, where b is in the column space of A. If the iteration is not converging for the given b, then how is it behaving? Because A is symmetric, how does the range of A relate to its null space? Use the fact that for any vector x E R^n, and any subspace S R^n, there exists a unique decomposition x = xS + xS, where xs E S and xS E S , the orthogonal complement of S. The vector XS is the orthogonal projection of x onto S (see the notes). Use the above iteration to show how the difference between two consecutive iterates, relates to the difference between the previous two iterates, . For a convergent iteration, this difference will converge to zero, but what does it converge to if A is singular, based on your work on Part (a)? And how can this be used to ''fix'' the iteration?

Explanation / Answer

Let A be an nxn symnetric non-singular matrix. We want to
reduce A to the "diagonal" form M D Mt by congruences, where D
is a block diagonal matrix, each block being of order 1 or 2, &nd H
is unit lower triangular with H 0 if D 0 0

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