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Consider the quadratic form f : R 2 R (x 1 x 2 ) 4(x 1 + 3x 2 ) 2 + 3(2x 1 x
Consider the quadratic form
f : R2 R
(x1 x2) 4(x1 + 3x2)2 + 3(2x1 x2)2.
a) Find matrices P, D M2(R), with D diagonal, such that
f(x) = (PTx)TD(PTx)
for all x R2 . The matrix P need not be orthogonal. Hint: Begin by writing the expression 4(x1 + 3x2)2 + 3(2x1 x2)2 in the form 1y12 + 2y22 for some new variables y1, y2. How are the vectors (x1, x2) and (y1, y2) related?
b) If A M2(R) is the symmetric matrix associated to f, show that A = P DP T. You may use the fact that if B, C Mn(R) are symmetric and xTBx = xTCx for all x Rn, then B = C.
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