W.D. Henshaw Math 6800: Solutions for Problem Set 4 1.Consider the Householder reflector, F=I-2uu*,u* u= 1. Determine the eigenvalues and eigenvectors, determinant, and singular values ofF.

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2.General Householder reflector. Letx, y∈Cm , withm >1. Show explicitly (using algebra) that ifkxk2=kyk2then there is Householder reflectorF=I-2uu* ,kuk2 = 1, such thatFx=αywhere α=±sign(y*x). Note: ifz=reiθ∈C, withr, θ∈Randr≥0 then sign(z) =eiθ , sign(0)≡1). (Hint: ifx6 =αyconsiderv=x-αy,u=v/kvk2 )

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Now v*v= (x-αy)*(x-αy) =x*x-¯αy*x-αx*y+y*y= 2(x*x-¯αy* x), v*x= (x-αy)*x=x*x-¯αy*x=1 2 v* v where we have usedkxk2=x*x=y*y=kyk2, ¯αα=|α|2= 1, and ¯αy*x=αx*y=±|x* y|since α=±sign(y*x). Thusv*x=1 2 (v* v) which proves the result. 3.Write a Matlab function[W,R] = house(A)that computes an implicit representation of a full or reduced QR factorization forA∈Cm×n withm≥nusing Householder reflections. The output variables are a lower triangular matrixW∈Cm×n whose columns are the Householder vectorsvk , and an upper triangular matrixR∈CRn×n . Also write a Matlab functionQ = formQ(W)that takes the matrixWfromhouseand generates the full matrixQ∈Cm×m . (a) Test your program on the Vandermonde matrix from problem set 3 withm= 5. CompareQ andRfrom the Matlab function[Q,R]=qr(A)to the output fromhouseandformQ. PrintQand R