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Problem_Set_5

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

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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 > 0 , q > 0 , p 2 > 4 q and ( p - q ) < 1 Denote the characteristic roots of A by λ 1 and λ 2 . (i) Show that λ 1 and λ 2 are real and positive. (ii) Show that exactly one of the following alternatives must occur: (A) λ 1 < 1 and λ 2 < 1; (B) λ 1 > 1 and λ 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 λ 1 and λ 2 . (a) Show that there is b = ( b 1 , b 2 ) R 2 , with b 6 = 0 , such that ( A - λ 1 I ) b = 0 . This shows that there is a real eigenvector corresponding to the eigenvalue λ 1 . (b) Define y = ( y 1 , y 2 ) as follows: y 1 = b 1 + ib 1 , y 2 = b 2 + ib 2 . Is y also an eigenvector of A (corresponding to the eigenvalue λ 1 )? Explain.

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

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