Take from Sections 15.7-8-9

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Goal
Given a quadratic form, find out if it has a minimum, or a
maximum, or neither

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Question
Can we classify the quadratic form
f (x, y ) = ax 2 + 2bxy + cy 2
by looking at a, b, and c?

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. 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. 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. Connection with matrices
Notice that
ab x
ax 2 + 2bxy + cy 2 = x y
bc y
So quadratic forms correspond with symmetric matrices.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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.