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Take from Sections 15.7-8-9

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1. 1. Lesson 22 (Sections 15.7–9) Quadratic Forms Math 20 November 9, 2007 Announcements Problem Set 8 on the website. Due November 14. No class November 12. Yes class November 21. next OH: Tue 11/13 3–4, Wed 11/14 1–3 (SC 323) next PS: Sunday? 6–7 (SC B-10), Tue 1–2 (SC 116)
2. 2. Outline Algebra primer: Completing the square A discriminating monopolist Quadratic Forms in two variables Classiﬁcation of quadratic forms in two variables Brute Force Eigenvalues Classiﬁcation of quadratic forms in several variables
3. 3. Algebra primer: Completing the square Remember that b aX 2 + bX + c = a X 2 + X +c a 2 2 b b − =a X+ +c 2a 2a 2 b2 b +c − =a X+ 2a 4a
4. 4. Algebra primer: Completing the square Remember that b aX 2 + bX + c = a X 2 + X +c a 2 2 b b − =a X+ +c 2a 2a 2 b2 b +c − =a X+ 2a 4a If a > 0, the function is an upwards-opening parabola and has b2 minimum value c − 4a If a < 0, the function is a downwards-opening parabola and b2 has maximum value c − 4a
5. 5. Outline Algebra primer: Completing the square A discriminating monopolist Quadratic Forms in two variables Classiﬁcation of quadratic forms in two variables Brute Force Eigenvalues Classiﬁcation of quadratic forms in several variables
6. 6. Example A ﬁrm sells a product in two separate areas with distinct linear demand curves, and has monopoly power to decide how much to sell in each area. How does its maximal proﬁt depend on the demand in each area?
7. 7. Example A ﬁrm sells a product in two separate areas with distinct linear demand curves, and has monopoly power to decide how much to sell in each area. How does its maximal proﬁt depend on the demand in each area? Let the demand curves be given by P1 = a1 − b1 Q1 P2 = a2 − b2 Q2 And the cost function by C = α(Q1 + Q2 ). The proﬁt is therefore π = P1 Q1 + P2 Q2 − α(Q1 + Q2 ) = (a1 − b1 Q1 )Q1 + (a2 − b2 Q2 )Q2 − α(Q1 + Q2 ) 2 2 = (a1 − α)Q1 − b1 Q1 + (a2 − α)Q2 − b2 Q2
8. 8. Solution Completing the square gives 2 2 π = (a1 − α)Q1 − b1 Q1 + (a2 − α)Q2 − b2 Q2 2 (a1 − α)2 (a1 − α) = −b1 Q1 − + 2b1 4b1 2 (a2 − α)2 (a2 − α) − b2 Q 2 − + 2b2 4b2 The optimal quantities are a1 − α a2 − α ∗ ∗ Q1 = Q2 = 2b1 2b2
9. 9. The corresponding prices are a1 + α a2 + α ∗ ∗ P1 = P2 = 2 2 The maximum proﬁt is (a1 − α)2 (a2 − α)2 π∗ = + 4b1 4b2
10. 10. Outline Algebra primer: Completing the square A discriminating monopolist Quadratic Forms in two variables Classiﬁcation of quadratic forms in two variables Brute Force Eigenvalues Classiﬁcation of quadratic forms in several variables
11. 11. Quadratic Forms in two variables Deﬁnition A quadratic form in two variables is a function of the form f (x, y ) = ax 2 + 2bxy + cy 2
12. 12. Quadratic Forms in two variables Deﬁnition A quadratic form in two variables is a function of the form f (x, y ) = ax 2 + 2bxy + cy 2 Example f (x, y ) = x 2 + y 2
13. 13. Quadratic Forms in two variables Deﬁnition A quadratic form in two variables is a function of the form f (x, y ) = ax 2 + 2bxy + cy 2 Example f (x, y ) = x 2 + y 2 f (x, y ) = −x 2 − y 2
14. 14. Quadratic Forms in two variables Deﬁnition A quadratic form in two variables is a function of the form f (x, y ) = ax 2 + 2bxy + cy 2 Example f (x, y ) = x 2 + y 2 f (x, y ) = −x 2 − y 2 f (x, y ) = x 2 − y 2
15. 15. Quadratic Forms in two variables Deﬁnition A quadratic form in two variables is a function of the form f (x, y ) = ax 2 + 2bxy + cy 2 Example f (x, y ) = x 2 + y 2 f (x, y ) = −x 2 − y 2 f (x, y ) = x 2 − y 2 f (x, y ) = 2xy
16. 16. Goal Given a quadratic form, ﬁnd out if it has a minimum, or a maximum, or neither
17. 17. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0
18. 18. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing.
19. 19. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is
20. 20. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite
21. 21. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is
22. 22. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is negative deﬁnite
23. 23. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is negative deﬁnite f (x, y ) = x 2 − y 2 is
24. 24. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is negative deﬁnite f (x, y ) = x 2 − y 2 is indeﬁnite
25. 25. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is negative deﬁnite f (x, y ) = x 2 − y 2 is indeﬁnite f (x, y ) = 2xy is
26. 26. Classes of quadratic forms Deﬁnition Let f (x, y ) be a quadratic form. f is said to be positive deﬁnite if f (x, y ) > 0 for all (x, y ) = (0, 0). f is said to be negative deﬁnite if f (x, y ) < 0 for all (x, y ) = (0, 0). f is said to be indeﬁnite if there exists points (x + , y + ) and (x − , y − ) such that f (x + , y + ) > 0 and f (x − , y − ) < 0 Example Classify these by inspection or by graphing. f (x, y ) = x 2 + y 2 is positive deﬁnite f (x, y ) = −x 2 − y 2 is negative deﬁnite f (x, y ) = x 2 − y 2 is indeﬁnite f (x, y ) = 2xy is indeﬁnite
27. 27. f (x, y ) class shape zero is a x2 + y2 positive upward- minimum deﬁnite opening paraboloid −x 2 − y 2 negative downward- maximum deﬁnite opening paraboloid x2 − y2 indeﬁnite saddle neither 2xy indeﬁnite saddle neither
28. 28. Notice that our discriminating monopolist objective function started out as a polynomial in two variables, and ended up the sum of a quadratic form and a constant. This is true in general, so when looking for extreme values, we can classify the associated quadratic form.
29. 29. Question Can we classify the quadratic form f (x, y ) = ax 2 + 2bxy + cy 2 by looking at a, b, and c?
30. 30. Outline Algebra primer: Completing the square A discriminating monopolist Quadratic Forms in two variables Classiﬁcation of quadratic forms in two variables Brute Force Eigenvalues Classiﬁcation of quadratic forms in several variables
31. 31. Brute Force Complete the square! f (x, y ) = ax 2 + 2bxy + cy 2 2 b2 y 2 by + cy 2 − =a x+ a a 2 ac − b 2 2 by =a x+ + y a a
32. 32. Brute Force Complete the square! f (x, y ) = ax 2 + 2bxy + cy 2 2 b2 y 2 by + cy 2 − =a x+ a a 2 ac − b 2 2 by =a x+ + y a a Fact Let f (x, y ) = ax 2 + 2bxy + cy 2 be a quadratic form. If a > 0 and ac − b 2 > 0, then f is positive deﬁnite If a < 0 and ac − b 2 > 0, then f is negative deﬁnite If ac − b 2 < 0, then f is indeﬁnite
33. 33. Connection with matrices Notice that ab x ax 2 + 2bxy + cy 2 = x y bc y So quadratic forms correspond with symmetric matrices.
34. 34. Eigenvalues Recall: Theorem (Spectral Theorem for Symmetric Matrices) Suppose An×n is symmetric, that is, A = A. Then A is diagonalizable. In fact, the eigenvectors can be chosen to be pairwise orthogonal with length one, which means that P−1 = P . Thus a symmetric matrix can be diagonalized as A = PDP , where D is diagonal and PP = In .
35. 35. So there exist numbers α, β, γ, δ such that ab αβ λ1 0 αγ = bc γδ 0 λ2 βδ Thus αβ λ1 0 αγ x f (x, y ) = x y γδ 0 λ2 βδ y λ1 0 αx + γy = αx + γy βx + δy 0 λ2 βx + δy = λ1 (αx + γy )2 + λ2 (βx + δy )2
36. 36. Upshot Fact ab Let f (x, y ) = ax 2 + 2bxy + cy 2 , and A = . Then: bc f is positive deﬁnite if and only if the eigenvalues of A ore positive f is negative deﬁnite if and only if the eigenvalues of A are negative f is indeﬁnite if one eigenvalue of A is positive and one is negative
37. 37. Outline Algebra primer: Completing the square A discriminating monopolist Quadratic Forms in two variables Classiﬁcation of quadratic forms in two variables Brute Force Eigenvalues Classiﬁcation of quadratic forms in several variables
38. 38. Classiﬁcation of quadratic forms in several variables Deﬁnition A quadratic form in n variables is a function of the form n Q(x1 , x2 , . . . , xn ) = aij xi xj i,j=1 where aij = aji .
39. 39. Classiﬁcation of quadratic forms in several variables Deﬁnition A quadratic form in n variables is a function of the form n Q(x1 , x2 , . . . , xn ) = aij xi xj i,j=1 where aij = aji . Q corresponds to the matrix A = (aij )n×n in the sense that Q(x) = x Ax
40. 40. Classiﬁcation of quadratic forms in several variables Deﬁnition A quadratic form in n variables is a function of the form n Q(x1 , x2 , . . . , xn ) = aij xi xj i,j=1 where aij = aji . Q corresponds to the matrix A = (aij )n×n in the sense that Q(x) = x Ax Deﬁnitions of positive deﬁnite, negative deﬁnite, and indeﬁnite go over mutatis mutandis.
41. 41. Theorem Let Q be a quadratic form, and A the symmetric matrix associated to Q. Then Q is positive deﬁnite if and only if all eigenvalues of A are positive Q is negative deﬁnite if and only if all eigenvalues of A are negative Q is indeﬁnite if and only if at least two eigenvalues of A have opposite signs.
42. 42. Theorem Let Q be a quadratic form, and A the symmetric matrix associated to Q. For each i = 1, . . . , n, let Di be the ith principal minor of A. Then Q is positive deﬁnite if and only if Di > 0 for all i Q is negative deﬁnite if and only if (−1)i Di > 0 for all i; that is, if and only if the signs of Di alternate and start negative.
43. 43. Theorem Let Q be a quadratic form, and A the symmetric matrix associated to Q. For each i = 1, . . . , n, let Di be the ith principal minor of A. Then Q is positive deﬁnite if and only if Di > 0 for all i Q is negative deﬁnite if and only if (−1)i Di > 0 for all i; that is, if and only if the signs of Di alternate and start negative. The proof is messy, but makes sense for diagonal A.