Lesson 22: Quadratic Forms

Take from Sections 15.7-8-9

1. Lesson 22 (Sections 15.7{9)
Quadratic Forms
Math 20
November 9, 2007
Announcements
IProblem Set 8 on the website. Due November 14.
INo class November 12. Yes class November 21.
Inext OH: Tue 11/13 3{4, Wed 11/14 1{3 (SC 323)
Inext 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
Classication of quadratic forms in two variables
Brute Force
Eigenvalues
Classication of quadratic forms in several variables

3. Algebra primer: Completing the square
Remember that
aX
2
+bX+c=a

X
2
+
b
a
X

+c
=a
"

X+
b
2a

2


b
2a

2
#
+c
=a

X+
b
2a

2
+c
b
2
4a
IIfa>0, the function is anupwards-opening parabola and has
minimumvaluec
b
2
4a
IIfa<0, the function is adownwards-opening parabola and
hasmaximumvaluec
b
2
4a

4. Algebra primer: Completing the square
Remember that
aX
2
+bX+c=a

X
2
+
b
a
X

+c
=a
"

X+
b
2a

2


b
2a

2
#
+c
=a

X+
b
2a

2
+c
b
2
4a
IIfa>0, the function is anupwards-opening parabola and has
minimumvaluec
b
2
4a
IIfa<0, the function is adownwards-opening parabola and
hasmaximumvaluec
b
2
4a

5. Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classication of quadratic forms in two variables
Brute Force
Eigenvalues
Classication of quadratic forms in several variables

6. Example
A rm 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 prot depend on the
demand in each area?
Let the demand curves be given by
P1=a1b1Q1 P2=a2b2Q2
And the cost function byC=(Q1+Q2). The prot is therefore
=P1Q1+P2Q2(Q1+Q2)
= (a1b1Q1)Q1+ (a2b2Q2)Q2(Q1+Q2)
= (a1)Q1b1Q
2
1+ (a2)Q2b2Q
2
2

7. Example
A rm 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 prot depend on the
demand in each area?
Let the demand curves be given by
P1=a1b1Q1 P2=a2b2Q2
And the cost function byC=(Q1+Q2). The prot is therefore
=P1Q1+P2Q2(Q1+Q2)
= (a1b1Q1)Q1+ (a2b2Q2)Q2(Q1+Q2)
= (a1)Q1b1Q
2
1+ (a2)Q2b2Q
2
2

8. Solution
Completing the square gives
= (a1)Q1b1Q
2
1+ (a2)Q2b2Q
2
2
=b1

Q1
(a1)
2b1

2
+
(a1)
2
4b1
b2

Q2
(a2)
2b2

2
+
(a2)
2
4b2
The optimal quantities are
Q

1=
a1
2b1
Q

2=
a2
2b2

9. The corresponding prices are
P

1=
a1+
2
P

2=
a2+
2
The maximum prot is


=
(a1)
2
4b1
+
(a2)
2
4b2

10. Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classication of quadratic forms in two variables
Brute Force
Eigenvalues
Classication of quadratic forms in several variables

11. Quadratic Forms in two variables
Denition
Aquadratic formin two variables is a function of the form
f(x;y) =ax
2
+ 2bxy+cy
2
Example
If(x;y) =x
2
+y
2
If(x;y) =x
2
y
2
If(x;y) =x
2
y
2
If(x;y) = 2xy

12. Quadratic Forms in two variables
Denition
Aquadratic formin two variables is a function of the form
f(x;y) =ax
2
+ 2bxy+cy
2
Example
If(x;y) =x
2
+y
2
If(x;y) =x
2
y
2
If(x;y) =x
2
y
2
If(x;y) = 2xy

13. Quadratic Forms in two variables
Denition
Aquadratic formin two variables is a function of the form
f(x;y) =ax
2
+ 2bxy+cy
2
Example
If(x;y) =x
2
+y
2
If(x;y) =x
2
y
2
If(x;y) =x
2
y
2
If(x;y) = 2xy

14. Quadratic Forms in two variables
Denition
Aquadratic formin two variables is a function of the form
f(x;y) =ax
2
+ 2bxy+cy
2
Example
If(x;y) =x
2
+y
2
If(x;y) =x
2
y
2
If(x;y) =x
2
y
2
If(x;y) = 2xy

15. Quadratic Forms in two variables
Denition
Aquadratic formin two variables is a function of the form
f(x;y) =ax
2
+ 2bxy+cy
2
Example
If(x;y) =x
2
+y
2
If(x;y) =x
2
y
2
If(x;y) =x
2
y
2
If(x;y) = 2xy

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

17. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

18. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

19. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

20. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

21. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

22. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

23. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

24. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

25. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

26. Classes of quadratic forms
Denition
Letf(x;y) be a quadratic form.
Ifis said to bepositive deniteiff(x;y)>0 for all
(x;y)6= (0;0).
Ifis said to benegative deniteiff(x;y)<0 for all
(x;y)6= (0;0).
Ifis said to beindeniteif there exists points (x
+
;y
+
) and
(x

;y

) such thatf(x
+
;y
+
)>0 andf(x

;y

)<0
Example
Classify these by inspection or by graphing.
If(x;y) =x
2
+y
2
is
positive denite
If(x;y) =x
2
y
2
is
negative denite
If(x;y) =x
2
y
2
is
indenite
If(x;y) = 2xyis
indenite

27. f(x;y) class shape zero is a
x
2
+y
2
positive
denite
upward-
opening
paraboloid
minimum
x
2
y
2
negative
denite
downward-
opening
paraboloid
maximum
x
2
y
2
indenite saddle neither
2xy indenite 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
Classication of quadratic forms in two variables
Brute Force
Eigenvalues
Classication of quadratic forms in several variables

31. Brute Force
Complete the square!
f(x;y) =ax
2
+ 2bxy+cy
2
=a

x+
by
a

2
+cy
2

b
2
y
2
a
=a

x+
by
a

2
+
acb
2
a
y
2
Fact
Let f(x;y) =ax
2
+ 2bxy+cy
2
be a quadratic form.
IIf a>0and acb
2
>0, then f is positive denite
IIf a<0and acb
2
>0, then f is negative denite
IIf acb
2
<0, then f is indenite

32. Brute Force
Complete the square!
f(x;y) =ax
2
+ 2bxy+cy
2
=a

x+
by
a

2
+cy
2

b
2
y
2
a
=a

x+
by
a

2
+
acb
2
a
y
2
Fact
Let f(x;y) =ax
2
+ 2bxy+cy
2
be a quadratic form.
IIf a>0and acb
2
>0, then f is positive denite
IIf a<0and acb
2
>0, then f is negative denite
IIf acb
2
<0, then f is indenite

33. Connection with matrices
Notice that
ax
2
+ 2bxy+cy
2
=

x y


a b
b c

x
y

So quadratic forms correspond with symmetric matrices.

34. Eigenvalues
Recall:
Theorem (Spectral Theorem for Symmetric Matrices)
SupposeAnnis symmetric, that is,A
0
=A. ThenAis
diagonalizable. In fact, the eigenvectors can be chosen to be
pairwise orthogonal with length one, which means thatP
1
=P
0
.
Thus a symmetric matrix can be diagonalized as
A=PDP
0
;
whereDis diagonal andPP
0
=In.

35. So there exist numbers,,,such that

a b
b c

=




10
02




Thus
f(x;y) =

x y





10
02




x
y

=

x+yx+y


10
02

x+y
x+y

=1(x+y)
2
+2(x+y)
2

36. Upshot
Fact
Let f(x;y) =ax
2
+ 2bxy+cy
2
, andA=

a b
b c

. Then:
If is positive denite if and only if the eigenvalues ofAore
positive
If is negative denite if and only if the eigenvalues ofAare
negative
If is indenite if one eigenvalue ofAis positive and one is
negative

37. Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classication of quadratic forms in two variables
Brute Force
Eigenvalues
Classication of quadratic forms in several variables

38. Classication of quadratic forms in several variables
Denition
Aquadratic forminnvariables is a function of the form
Q(x1;x2; : : : ;xn) =
n
X
i;j=1
aijxixj
whereaij=aji.
Qcorresponds to the matrixA= (aij)nnin the sense that
Q(x) =x
0
Ax
Denitions of positive denite, negative denite, and indenite go
overmutatis mutandis.

39. Classication of quadratic forms in several variables
Denition
Aquadratic forminnvariables is a function of the form
Q(x1;x2; : : : ;xn) =
n
X
i;j=1
aijxixj
whereaij=aji.
Qcorresponds to the matrixA= (aij)nnin the sense that
Q(x) =x
0
Ax
Denitions of positive denite, negative denite, and indenite go
overmutatis mutandis.

40. Classication of quadratic forms in several variables
Denition
Aquadratic forminnvariables is a function of the form
Q(x1;x2; : : : ;xn) =
n
X
i;j=1
aijxixj
whereaij=aji.
Qcorresponds to the matrixA= (aij)nnin the sense that
Q(x) =x
0
Ax
Denitions of positive denite, negative denite, and indenite go
overmutatis mutandis.

41. Theorem
Let Q be a quadratic form, and A the symmetric matrix associated
to Q. Then
IQ is positive denite if and only if all eigenvalues of A are
positive
IQ is negative denite if and only if all eigenvalues of A are
negative
IQ is indenite 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 Dibe the ith principal minor ofA.
Then
IQ is positive denite if and only if Di>0for all i
IQ is negative denite if and only if(1)
i
Di>0for all i; that
is, if and only if the signs of Dialternate and start negative.
The proof is messy, but makes sense for diagonalA.

43. Theorem
Let Q be a quadratic form, and A the symmetric matrix associated
to Q. For each i= 1; : : : ;n, let Dibe the ith principal minor ofA.
Then
IQ is positive denite if and only if Di>0for all i
IQ is negative denite if and only if(1)
i
Di>0for all i; that
is, if and only if the signs of Dialternate and start negative.
The proof is messy, but makes sense for diagonalA.