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slide8-sept9 - ECON 401: Quadratic forms and semidenite...

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matrices Siyang Xiong Rice University September 11, 2011 Xiong (Rice University) ECON 401 September 11, 2011 1 / 22
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a bijection : f 1 , 2 , ..., n g ! f 1 , 2 , ..., n g is called a permutation of f 1 , 2 , ..., n g . for example: [ 1 , 3 , 2 ] is a permutation of f 1 , 2 , 3 g [ 1 , 4 , 2 , 5 , 3 ] is a permutation of f 1 , 2 , 3 , 4 , 5 g for f 1 , 2 g , the permutations are [ 1 , 2 ] and [ 2 , 1 ] for f 1 , 2 , 3 g , the permutations are [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] , [ 2 , 1 , 3 ] , [ 2 , 3 , 1 ] , [ 3 , 1 , 2 ] , [ 3 , 2 , 1 ] . Xiong (Rice University) ECON 401 September 11, 2011 2 / 22
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Given a n n symmetric matrix A and a permutation of f 1 , 2 , ..., n g , , A D 2 4 a 1 1 ... a 1 n ... a i j ... a n 1 ... a n n 3 5 , for example: for A D a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ± and D [ 2 , 1 ] : A D a 2 , 2 a 2 , 1 a 1 , 2 a 1 , 1 ± . Xiong (Rice University) ECON 401 September 11, 2011 3 / 22
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for example: for A D 2 4 a 1 , 1 a 1 , 2 a 1 , 3 a 2 , 1 a 2 , 2 a 2 , 3 a 3 , 1 a 3 , 2 a 3 , 3 3 5 and D [ 2 , 1 , 3 ] : A D 2 4 a 2 , 2 a 2 , 1 a 2 , 3 a 1 , 2 a 1 , 1 a 1 , 3 a 3 , 2 a 3 , 1 a 3 , 3 3 5 . Xiong (Rice University) ECON 401 September 11, 2011 4 / 22
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Theorem: A A k 0 for all and all k D 1 , 2 , ..., n . Theorem: A . ± 1 / k A k 0 for all and all k D 1 , 2 , ..., n . Xiong (Rice University) ECON 401 September 11, 2011 5 / 22
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consider A D 1 0 0 0 ± , there are two permutations: D [ 1 , 2 ] and & 0 D [ 2 , 1 ] A D 1 0 0 0 ± and A & 0 D & 0 0 0 1 ± . ² ² A 1 ² ² D ² ² [ 1 ] ² ² D 1 & 0 ² ² A 2 ² ² D ² ² ² ² 1 0 0 0 ±² ² ² ² D 0 & 0 ² ² ² A & 0 1 ² ² ² D ² ² [ 0 ] ² ² D 0 & 0 ² ² ² A & 0 2 ² ² ² D ² ² ² ² & 0 0 0 1 ±² ² ² ² D 0 & 0 Therefore, 1 0 0 0 ±
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