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7.3 EXERCISES In Exercises 1 and 2, find the change of variable x = Py that In Exercises 3-6, find (a) the maximum value of Q(x) subject to transforms the quadratic form x'Ax into y' Dy as shown. the constraint x x = 1, (b) a unit vector u where this maximum is 1. 5x} +6x; +7x}+4x1X2-4x2.*3 =971+6y2+3y} attained, and (c) the maximum of O(x) subject to the constraints x x = 1 and xu = 0. 2. 3.x7+ 3x3 +5x] +6x1x2+2xj13+2x213=7y7+ 4y Hint: x and y must have the same number of coordinates, so the 3. Q(x) = 5x7 + 6x} + 7x3 + 4x1-*2 -4x2-X3 quadratic form shown here must have a coefficient of zero for y;. (See Exercise 1.) SECOND REVISED PAGES aptara 416 CHAPTER 7 Symmetric Matrices and Quadratic Forms A. Q(x) = 3x7 +3.x3+5x3 +6x112+2xx3+2x2x; (See Exer- 12, Let A be any eigenvalue of a symmetric matrix A. Justify cise 2.) the statement made in this section that m