The Rayleigh quotient iteration

[math8]

I am trying to apply the Rayleigh quotient iteration to the matrix A=

a 0
0 b

where a is different from b (a and b are reals).

Find the subsets S of R^2(Reals^2) having the property that the iteration applied to this matrix with initial guesses in S do not converge. Is S a set of measure 0?

I know the Rayleigh quotient is R(A,x)=x*Ax/x*x for A Hermitian and x nonzero. I am supposed to begin with an eigenvalue guess mo and an eigenvector guess vo and use the R.Q. but I am not sure which guesses should I begin with.

I am not sure how to proceed here. Any help is much appreciated.

 

[kurt.physics]

The set S of initial guesses which do not converge is the set of vectors whose entries are both 0 or both non-zero. This set is a measure 0 set, as it has zero area in the plane. To apply the Rayleigh quotient iteration for this matrix, we start with an initial guess (m_0, v_0) such that m_0 is a guessed eigenvalue and v_0 is a corresponding guessed eigenvector. We then iterate by computing the Rayleigh quotient at each step: m_(k+1) = v_k*A*v_k/(v_k*v_k) v_(k+1) = A*v_k/|A*v_k|We then repeat this process until convergence. Note that if the initial guess (m_0, v_0) satisfies the condition that both entries of v_0 are either 0 or both non-zero, then this iteration will not converge, since the magnitude of A*v_k will always be 0.