Lecture 5 Principal Minors and the Hessian

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Optimization of functions in several variables The Hessian matrix Let f (x) be a function in n variables. The Hessian matrix of f is the matrix consisting of all the second order partial derivatives of f : Definition The Hessian matrix of f at the point x is the n × n matrix ⎛ f 11 ′′ (x) f ′′ 12 (x) . . . f ′′ 1n (x) ⎞ f ′′ f 21 ′′ (x) f ′′ 22 (x) . . . f ′′ 2n (x) = ⎜ (x) ⎝ . . . .. ⎟ . ⎠ f n1 ′′ (x) f ′′ n2 (x) . . . f ′′ nn(x) Notice that each entry in the Hessian matrix is a second order partial derivative, and therefore a function in x. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, 2010 12 / 25
Optimization of functions in several variables The Hessian matrix: An example Example Compute the Hessian matrix of the quadratic form f (x 1 , x 2 ) = x 2 1 − x 2 2 − x 1 x 2 Solution We computed the first order partial derivatives above, and found that Hence we get f ′ 1(x 1 , x 2 ) = 2x 1 − x 2 , f ′ 2(x 1 , x 2 ) = −2x 2 − x 1 f 11(x ′′ 1 , x 2 ) = 2 f 12(x ′′ 1 , x 2 ) = −1 f 21(x ′′ 1 , x 2 ) = −1 f 22(x ′′ 1 , x 2 ) = −2 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, 2010 13 / 25
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Lecture 5 Principal Minors and the Hessian

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