204PS52008 - Econ 204, Summer/Fall 2008 Problem Set 5 Due...

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Econ 204, Summer/Fall 2008 Problem Set 5 Due in Lecture Friday, August 15 1. In the following cases, show that matrices A and B have the same characteristic polynomials and hence the same eigenvalues. (a) B = A > , the transpose of A . (b) A and B are similar matrices. 2. Show that if an invertible matrix A is diagonalizable, then A 1 is diagonalizable. 3. Identify which of the following matrices can be diagonalized and provide the diagonalization. If you claim that a diagonalization does not exist, prove it: (a) 21 23 (b) 3 1 03 (c) 41 1 24 1 4 30 (d) 32 1 14 1 12 1 4. Consider the following quadratic forms: f ( x,y )=6 x 2 +4 xy +3 y 2 ,g ( xy Answer the following questions for each of these forms: (a) Find a symmetric matrix M such that the form equals ( xy ) M ( x y ) . (b) Find the eigenvalues of the form. (c) Find an orthonormal basis of eigenvectors.
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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204PS52008 - Econ 204, Summer/Fall 2008 Problem Set 5 Due...

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