Problem_Set_5 - T.Mitra, Fall, 2008 Economics 6170 Problem...

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Unformatted text preview: T.Mitra, Fall, 2008 Economics 6170 Problem Set 5 [Due on Wednesday, October 15] 1. [Non-Symmetric Matrices and Real Eigenvalues] Let A be the 2 2 matrix, given by: A = a 11 a 12 a 21 a 22 where a 12 6 = a 21 . Let p be the trace of A, and let q be the determinant of A. Assume that: p > ,q > , p 2 > 4 q and ( p- q ) < 1 Denote the characteristic roots of A by 1 and 2 . (i) Show that 1 and 2 are real and positive. (ii) Show that exactly one of the following alternatives must occur: (A) 1 < 1 and 2 < 1; (B) 1 > 1 and 2 > 1 . 2. [Eigenvalues and Eigenvectors of Symmetric Matrices] Let A = ( a ij ) be a symmetric 2 2 matrix. We know that it has only real eigenvalues; denote these by 1 and 2 . (a) Show that there is b = ( b 1 ,b 2 ) R 2 , with b 6 = 0 , such that ( A- 1 I ) b = . This shows that there is a real eigenvector corresponding to the eigenvalue 1 ....
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This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell University (Engineering School).

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Problem_Set_5 - T.Mitra, Fall, 2008 Economics 6170 Problem...

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