Let ARmn,bRm,<,>:RmRmR be some inner product in Rm and 2=<,>

the corresponding norm. Consider the least-squares problem: find xRn such that

Axb2

is minimized.

a) By Lemma (1) there exists a symmetric and positive definite matrix MRmm such that <y,z>=zTMy

for all y,zRm. Decompose M=QQT. Show that each eigenvalue of M is positive and find a matrix

LRmm such that M=LLT.

b) Use (a) to reformulate the above problem as a least-squares problem posed in the Euclidian norm. Tht is find

A~Rmn and b~Rm such that

Axb2=A~xb~22 for all xRn.

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Lemma (1): For any inner product <,>:RnRnR, there exists a symmetrix positive definite matrix

ARnn such that

<x,y>=yTAx. (21)

for all xRn. On the other hand the formula (21) defines an inner product for any symmetric positive

definite ARnn.

Hint: Give the matrix A~ and vector b~ as a product of A,b , and L. You do not have to define L explicitly, it may remain

as a part of the solution.