(c) The QR iteration with shifts for computing the eigenvalues of a matrix A Rx takes...

  

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(c) The QR iteration with shifts for computing the eigenvalues of a matrix A Rx takes the form 1 Ho = UT AU (initial reduction to upper Hessenberg form) 2 for k = 1, 2,... 3 - HIQkRk (QR factorization, possibly complex) 4 Hk+1:= RkQk + MkI 5 end (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] (c) The QR iteration with shifts for computing the eigenvalues of a matrix A Rx takes the form 1 Ho = UT AU (initial reduction to upper Hessenberg form) 2 for k = 1, 2,... 3 - HIQkRk (QR factorization, possibly complex) 4 Hk+1:= RkQk + MkI 5 end (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]