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slide7-sept7 - ECON 401: Quadratic forms and...

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Unformatted text preview: ECON 401: Quadratic forms and semide&nite matrices Siyang Xiong Rice University September 8, 2011 Xiong (Rice University) ECON 401 September 8, 2011 1 / 15 unconstrained maximization Throughout the lecture, we consider f : R n ! R . Consider the optimization problem max f . x / s.t. x 2 R n Xiong (Rice University) ECON 401 September 8, 2011 2 / 15 whether a critical point is a local maximizer or a local minimizer depends on D 2 f . x / Recall f : R n ! R ; D 2 f . x / is a n & n matrix, D 2 f . x / D 2 6 6 4 @ 2 f . x / @ x 1 @ x 1 ... @ 2 f . x / @ x 1 @ x n ... @ 2 f . x / @ x i @ x j ... @ 2 f . x / @ x n @ x 1 ... @ 2 f . x / @ x n @ x n 3 7 7 5 this is a symmetric matrix. Xiong (Rice University) ECON 401 September 8, 2011 3 / 15 quadratic forms consider a n & n symmetric matrix A , A D 2 4 a 11 ... a 1 n ... a ij ... a n 1 ... a nn 3 5 , i.e., a ij D a ji , or equivalently A D A . A de&nes a quadratic form: 3 A : R n ! R , for x D [ x 1 , ..., x n ] 2 R n 3 A . x / D xAx D [ x 1 , ..., x n ] 2 4 a 11 ... a 1 n ... a ij ......
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slide7-sept7 - ECON 401: Quadratic forms and...

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