Find a change of variables that reduces the quadratic form to a sum or differenc

Question

Find a change of variables that reduces the quadratic form to a sum or difference of squares, and express the quadratic form in terms of the new variables. 13x21 + 12x22 + 11x23 - 4X1X2 + 4x2x3 A substitution x = Py that eliminates cross-product terms is x1 = - y1 + 2y2 - 2y3, x2 = - 2y1 + y2 + 2y3, x3 = 2y1 + 2y2 + y3. The new quadratic form is 9 y21 + 12y22 + 15y23. A substitution x = Py that eliminates cross-product terms is X1 = - 1/3y1 + 2/3y2 - 2/3y3, X2 = - 2/3y1 + 1/3y2 + 2/3y3, X3 = - 2/3y1 + 2/3y2 - 1/3y3. The new quadratic form is 9y21 - 12y22 + 15 y23. A substitution x = Py that eliminates cross-product terms is X1 = - 1/3y1 - 2/3y2 - 2/3y3, X2 = - 2/3y1 - 1/3y2 + 2/3y3, X3 = 2/3y1 + 2/3y2 + 1/3y3. The new quadratic form is 12 y21 + 11 y22 + 9y23. A substitution x = Py that eliminates cross-product terms is x1 = - y1 - 2y2 - 2y3, x2 = - 2y1 - y2 + 2y3, x3 = 2y1 + 2y2 + y3. The new quadratic form is 15y21 + 9y22 + 12y23. A substitution x = Py that eliminates cross-product terms is X1 = - 1/3y1 + 2/3y2 - 2/3y3, X2 = - 2/3y1 + 1/3y2 + 2/3y3, X3 = 2/3y1 + 2/3y2 + 1/3y3. The new quadratic form is 9y21 + 12 y22 + 15y23.

Explanation / Answer

3rd option

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