Lesson 22: Quadratic Forms

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


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


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






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







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



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



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



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



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






Example











Example


I f (x , y) = x2 + y2 is

positive definite









Example











Example



I f (x , y) = −x2 − y2 is

negative definite








Example











Example




I f (x , y) = x2 − y2 is

indefinite







Example











Example





I f (x , y) = 2xy is

indefinite






Example






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






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







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.



Q(x1, x2, . . . , xn) =n∑

i ,j=1

aijxixj

where aij = aji .


Q(x) = x′Ax




Q(x1, x2, . . . , xn) =n∑

i ,j=1

aijxixj

where aij = aji .


Q(x) = x′Ax


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.

Lesson 22: Quadratic Forms

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