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# sol66 - Partial Solution Set Leon 6.6 6.6.1 Find the matrix...

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Partial Solution Set, Leon § 6.6 6.6.1 Find the matrix associated with each of the quadratic forms. (a) 3 x 2 - 5 xy + y 2 (b) 2 x 2 + 3 y 2 + z 2 + xy - 2 xz + 3 yz (c) x 2 + 2 y 2 + z 2 + xy - 2 xz + 3 yz Solution : (a) A = 3 - 5 / 2 - 5 / 2 1 . (b) A = 2 1 / 2 - 1 1 / 2 3 3 / 2 - 1 3 / 2 1 . (c) A = 1 1 / 2 - 1 1 / 2 2 3 / 2 - 1 3 / 2 1 . 6.6.6 Which of the following matrices are positive definite? Negative definite? Indefinite? (a) A = 3 2 2 2 is positive definite. Since it is 2 × 2, any test will do. (b) A = 3 4 4 1 is indefinite. Its determinant is -13, so it has eigenvalues of both signs. (d) A = - 2 0 1 0 - 1 0 1 0 - 2 is negative definite, since - A is positive definite. 6.6.7 For each of the following functions, determine whether the given stationary point corre- sponds to a local minimum, a local maximum, or a saddle point. (d) f ( x, y ) = y x 2 + x y 2 + xy , (1 , 1). (e) f ( x, y, z ) = x 3 + xyz + y 2 - 3 x , (1 , 0 , 0). Solution : (d) The Hessian of f at the given stationary point is H = 6 - 3 - 3 6 , with eigenvalues 3 and 9; both are positive, so the stationary point corresponds to a minimum.

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(e) The Hessian of f
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sol66 - Partial Solution Set Leon 6.6 6.6.1 Find the matrix...

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