Consider the function P1 : R2 R defined as follows: for any x=(x1,x2)R2,

set xP12=31x12+x1x2+x22.

a) Find a symmetric matrix AR2x2 such that

xP12=xTAx, for all xR2.

b) Show that A above is positive definite.

c) Show that xP12:R2R is a norm.

HINTS for b): -------------------------------------------------------------------

see definition: A matrix ARnxn is called positive semidefinite if

xTAx=xAx0

for all xRn. If the second equality above holds only for x=0 , then A is called positive definite.

Use: min(A)x22xAxmax(A)x22,

where min,maxR are the smallest and the largest eigenvalues of A.

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HINTS for c):

Use Lemma:

For any inner product <,>:RnRnR, there exists a symmetric positive definite matrix ARxx such that

<x,y>=yTAx (1)

for all xRn. On the other hand, formula (1) defines an inner product for any symmetric positive definite ARnn.