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Take from Sections 15.7-8-9
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Lesson 22 (Sections 15.7–9)Quadratic Forms
Math 20
November 9, 2007
Announcements
I Problem Set 8 on the website. Due November 14.
I No class November 12. Yes class November 21.
I next OH: Tue 11/13 3–4, Wed 11/14 1–3 (SC 323)
I next PS: Sunday? 6–7 (SC B-10), Tue 1–2 (SC 116)
Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classification of quadratic forms in two variablesBrute ForceEigenvalues
Classification of quadratic forms in several variables
Algebra primer: Completing the square
Remember that
aX 2 + bX + c = a
(X 2 +
b
aX
)+ c
= a
[(X +
b
2a
)2
−(
b
2a
)2]
+ c
= a
(X +
b
2a
)2
+ c − b2
4a
I If a > 0, the function is an upwards-opening parabola and hasminimum value c − b2
4a
I If a < 0, the function is a downwards-opening parabola andhas maximum value c − b2
4a
Algebra primer: Completing the square
Remember that
aX 2 + bX + c = a
(X 2 +
b
aX
)+ c
= a
[(X +
b
2a
)2
−(
b
2a
)2]
+ c
= a
(X +
b
2a
)2
+ c − b2
4a
I If a > 0, the function is an upwards-opening parabola and hasminimum value c − b2
4a
I If a < 0, the function is a downwards-opening parabola andhas maximum value c − b2
4a
Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classification of quadratic forms in two variablesBrute ForceEigenvalues
Classification of quadratic forms in several variables
Example
A firm sells a product in two separate areas with distinct lineardemand curves, and has monopoly power to decide how much tosell in each area. How does its maximal profit depend on thedemand in each area?
Let the demand curves be given by
P1 = a1 − b1Q1 P2 = a2 − b2Q2
And the cost function by C = α(Q1 + Q2). The profit is therefore
π = P1Q1 + P2Q2 − α(Q1 + Q2)
= (a1 − b1Q1)Q1 + (a2 − b2Q2)Q2 − α(Q1 + Q2)
= (a1 − α)Q1 − b1Q21 + (a2 − α)Q2 − b2Q2
2
Example
A firm sells a product in two separate areas with distinct lineardemand curves, and has monopoly power to decide how much tosell in each area. How does its maximal profit depend on thedemand in each area?
Let the demand curves be given by
P1 = a1 − b1Q1 P2 = a2 − b2Q2
And the cost function by C = α(Q1 + Q2). The profit is therefore
π = P1Q1 + P2Q2 − α(Q1 + Q2)
= (a1 − b1Q1)Q1 + (a2 − b2Q2)Q2 − α(Q1 + Q2)
= (a1 − α)Q1 − b1Q21 + (a2 − α)Q2 − b2Q2
2
Solution
Completing the square gives
π = (a1 − α)Q1 − b1Q21 + (a2 − α)Q2 − b2Q2
2
= −b1
(Q1 −
(a1 − α)
2b1
)2
+(a1 − α)2
4b1
− b2
(Q2 −
(a2 − α)
2b2
)2
+(a2 − α)2
4b2
The optimal quantities are
Q∗1 =a1 − α
2b1Q∗2 =
a2 − α2b2
The corresponding prices are
P∗1 =a1 + α
2P∗2 =
a2 + α
2
The maximum profit is
π∗ =(a1 − α)2
4b1+
(a2 − α)2
4b2
Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classification of quadratic forms in two variablesBrute ForceEigenvalues
Classification of quadratic forms in several variables
Quadratic Forms in two variables
DefinitionA quadratic form in two variables is a function of the form
f (x , y) = ax2 + 2bxy + cy2
Example
I f (x , y) = x2 + y2
I f (x , y) = −x2 − y2
I f (x , y) = x2 − y2
I f (x , y) = 2xy
Quadratic Forms in two variables
DefinitionA quadratic form in two variables is a function of the form
f (x , y) = ax2 + 2bxy + cy2
Example
I f (x , y) = x2 + y2
I f (x , y) = −x2 − y2
I f (x , y) = x2 − y2
I f (x , y) = 2xy
Quadratic Forms in two variables
DefinitionA quadratic form in two variables is a function of the form
f (x , y) = ax2 + 2bxy + cy2
Example
I f (x , y) = x2 + y2
I f (x , y) = −x2 − y2
I f (x , y) = x2 − y2
I f (x , y) = 2xy
Quadratic Forms in two variables
DefinitionA quadratic form in two variables is a function of the form
f (x , y) = ax2 + 2bxy + cy2
Example
I f (x , y) = x2 + y2
I f (x , y) = −x2 − y2
I f (x , y) = x2 − y2
I f (x , y) = 2xy
Quadratic Forms in two variables
DefinitionA quadratic form in two variables is a function of the form
f (x , y) = ax2 + 2bxy + cy2
Example
I f (x , y) = x2 + y2
I f (x , y) = −x2 − y2
I f (x , y) = x2 − y2
I f (x , y) = 2xy
Goal
Given a quadratic form, find out if it has a minimum, or amaximum, or neither
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is
positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is
negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is
indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is
indefinite
Classes of quadratic forms
DefinitionLet f (x , y) be a quadratic form.
I f is said to be positive definite if f (x , y) > 0 for all(x , y) 6= (0, 0).
I f is said to be negative definite if f (x , y) < 0 for all(x , y) 6= (0, 0).
I 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.
I f (x , y) = x2 + y2 is positive definite
I f (x , y) = −x2 − y2 is negative definite
I f (x , y) = x2 − y2 is indefinite
I f (x , y) = 2xy is indefinite
f (x , y) class shape zero is a
x2 + y2 positivedefinite
upward-openingparaboloid
minimum
−x2 − y2 negativedefinite
downward-openingparaboloid
maximum
x2 − y2 indefinite saddle neither
2xy indefinite saddle neither
Notice that our discriminating monopolist objective functionstarted out as a polynomial in two variables, and ended up the sumof a quadratic form and a constant. This is true in general, sowhen looking for extreme values, we can classify the associatedquadratic form.
QuestionCan we classify the quadratic form
f (x , y) = ax2 + 2bxy + cy2
by looking at a, b, and c?
Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classification of quadratic forms in two variablesBrute ForceEigenvalues
Classification of quadratic forms in several variables
Brute Force
Complete the square!
f (x , y) = ax2 + 2bxy + cy2
= a
(x +
by
a
)2
+ cy2 − b2y2
a
= a
(x +
by
a
)2
+ac − b2
ay2
FactLet f (x , y) = ax2 + 2bxy + cy2 be a quadratic form.
I If a > 0 and ac − b2 > 0, then f is positive definite
I If a < 0 and ac − b2 > 0, then f is negative definite
I If ac − b2 < 0, then f is indefinite
Brute Force
Complete the square!
f (x , y) = ax2 + 2bxy + cy2
= a
(x +
by
a
)2
+ cy2 − b2y2
a
= a
(x +
by
a
)2
+ac − b2
ay2
FactLet f (x , y) = ax2 + 2bxy + cy2 be a quadratic form.
I If a > 0 and ac − b2 > 0, then f is positive definite
I If a < 0 and ac − b2 > 0, then f is negative definite
I If ac − b2 < 0, then f is indefinite
Connection with matrices
Notice that
ax2 + 2bxy + cy2 =(x y
)(a bb c
)(xy
)So quadratic forms correspond with symmetric matrices.
Eigenvalues
Recall:
Theorem (Spectral Theorem for Symmetric Matrices)
Suppose An×n is symmetric, that is, A′ = A. Then A isdiagonalizable. In fact, the eigenvectors can be chosen to bepairwise 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.
So there exist numbers α, β, γ, δ such that(a bb c
)=
(α βγ δ
)(λ1 00 λ2
)(α γβ δ
)Thus
f (x , y) =(x y
)(α βγ δ
)(λ1 00 λ2
)(α γβ δ
)(xy
)=(αx + γy βx + δy
)(λ1 00 λ2
)(αx + γyβx + δy
)= λ1 (αx + γy)2 + λ2 (βx + δy)2
Upshot
Fact
Let f (x , y) = ax2 + 2bxy + cy2, and A =
(a bb c
). Then:
I f is positive definite if and only if the eigenvalues of A orepositive
I f is negative definite if and only if the eigenvalues of A arenegative
I f is indefinite if one eigenvalue of A is positive and one isnegative
Outline
Algebra primer: Completing the square
A discriminating monopolist
Quadratic Forms in two variables
Classification of quadratic forms in two variablesBrute ForceEigenvalues
Classification of quadratic forms in several variables
Classification of quadratic forms in several variables
DefinitionA quadratic form in n variables is a function of the form
Q(x1, x2, . . . , xn) =n∑
i ,j=1
aijxixj
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 goover mutatis mutandis.
Classification of quadratic forms in several variables
DefinitionA quadratic form in n variables is a function of the form
Q(x1, x2, . . . , xn) =n∑
i ,j=1
aijxixj
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 goover mutatis mutandis.
Classification of quadratic forms in several variables
DefinitionA quadratic form in n variables is a function of the form
Q(x1, x2, . . . , xn) =n∑
i ,j=1
aijxixj
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 goover mutatis mutandis.
TheoremLet Q be a quadratic form, and A the symmetric matrix associatedto Q. Then
I Q is positive definite if and only if all eigenvalues of A arepositive
I Q is negative definite if and only if all eigenvalues of A arenegative
I Q is indefinite if and only if at least two eigenvalues of A haveopposite signs.
TheoremLet Q be a quadratic form, and A the symmetric matrix associatedto Q. For each i = 1, . . . , n, let Di be the ith principal minor of A.Then
I Q is positive definite if and only if Di > 0 for all i
I Q is negative definite if and only if (−1)iDi > 0 for all i ; thatis, if and only if the signs of Di alternate and start negative.
The proof is messy, but makes sense for diagonal A.
TheoremLet Q be a quadratic form, and A the symmetric matrix associatedto Q. For each i = 1, . . . , n, let Di be the ith principal minor of A.Then
I Q is positive definite if and only if Di > 0 for all i
I Q is negative definite if and only if (−1)iDi > 0 for all i ; thatis, if and only if the signs of Di alternate and start negative.
The proof is messy, but makes sense for diagonal A.
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