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27.1. Let A E Cmxm be given, not necessarily hermitian. Show that a number Z E C is a Rayleigh quotient of A if and only if it is a diagonal entry of Q*AQ for some unitary matrix Q. Thus Rayleigh quotients are just diagonal entries of matrices, once you transform orthogonally to the right coordinate system. From web page the Rayleigh quotient R(M, = x*Mx x* x 27.5. As mentioned in the text, inverse iteration depends on the solution of a system of equations that may be exceedingly ill-conditioned, with condition number on the order of Emachine We know that it is impossible in general to solve ill-conditioned systems accurately. Is this not a fatal flaw in the algorithm? Show as follows that the answer is no-that ill-conditioning is not a problem in inverse iteration. Suppose A is a real symmetric matrix with one eigenvalue much smaller than the others in absolute value (without loss of generality, we are taking H = 0). Suppose U is a vector with components in the directions of all the eigenvalues 41 Im of A, and suppose Aw = U is solved backward stably, yielding a computed vector w. Making use of the calculation on p. 95, show that although w may be far from w, /|||| will not be far from w / will.

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