Question: (c) The QR iteration with shifts for computing the eigenvalues of a matrix A Rnxn takes the form 1 Ho = UT AU (initial reduction

(c) The QR iteration with shifts for computing the eigenvalues of a matrix A Rnxn takes the form 1 Ho = UT AU (initial reduction to upper Hessenberg form) 2 for k = 1, 2, ... Hk MRI =:QkRk (QR factorization, Hk possibly complex) Hk+1:= R&Qk + MkI 5 end 3 4 = (i) Show that each iteration represents a similarity transformation. [2 marks] (ii) Define an upper Hessenberg matrix and explain why we begin by reducing A to upper Hessenberg form. [2 marks]

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