Exercise 3

Exercise Session 3, November 1st , 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva 1. Determine the de…niteness of Q = 3x21 + x22 + 6x23 2 3 2 1 2 2 2 5: 2. Let A be symmetric matrix 4 1 2 2 5

2x1 x2 + 2x1 x3 + 4x2 x3

(i) Show that the characteristic polynomial may be written in the form ( 1)2 ( 7):

(ii) Show that the vector equation Ax = x reduces to a single scalar equation. Hence, …nd eigenvectors x; y corresponding to the eigenvalue 1 which have the property that xT y = 0: (iii) Find an eigenvector z corresponding to the eigenvalue 7. (iv) Find a diagonal matrix D and an orthogonal matrix S such that S 1 AS = D: (v) Find A

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and de…ne spectral decomposition.

2 3. The consumption set of a consumer is C = (x; y) 2 R+ : x > x0 > 0; y > y 0 > 0 : Illustrate this set. Is it closed? Bounded? How would you interpret x0 and y0 ? 2 : p1 x + p2 y m where 4. A consumer’s budget set is B = (x; y) 2 R+ p1 ; p2 > 0 are prices and m > 0 is income. Illustrate this set. Is it closed? Bounded?

5. Prove that: (i) an open set does not contain its boundary points (ii) the intersection of two closed sets is closed.

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