002 LS 4 Quadratic Forms

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Lec 4: Mathematical EconomicsQuadratic Forms

Sugata Bag

Delhi School of Economics

5th September 2013

[SB ] (Delhi School of Economics) Introductory Math Econ 5th September 2013 1 / 10
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Introduction Bilinear & Quadrati Forms

Denition: Bilinear Form

DenitionA bilinear form on R n is a function L = L(x , y ) of two argumentsx , y 2R n which is linear in each argument, i.e. such that

1. L(x 1 + x 2,

y ) = L(x 1,

y ) + L(x 2,

y );2. L(x , y 1 + y 2) = L(x , y 1) + L(x , y 2).

Note (1) If x = ( x 1 , ..., x n )T and y = ( y 1 , ..., y n)T , a bilinear form canbe written as

L(x ,

y ) = n j , k = 1 a jk x k y j = ( Ax

,

y ) = y Ax where Ann is determined uniquely by the linear form L.Note (2): L(x , y ) is essentially an inner product.

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Quadratic Forms

FactA quadratic form in n variables x 1 , ..., x n is an expression

F = a jk x k x j = a11 x 1x 1 + a12x 1x 2 + .... + a1nx 1x n+a 21x 2x 1 + ... + a2nx 2x n+......

+ an1x nx 1 + ...

+ ann x nx n.

It is called a "quadratic" form because each term a ij x i x j contains either the square of a variable or the product of two dierent variables. A QF onR n is the "diagonal" of a bilinear form L, i.e. that any quadratic form Q is dened by Q [x ] = L(x , x ) = ( Ax , x ) = x Ax .

Examples

(1) A QF (a11 x 21 + 2a12x 1x 2 + a22x 22 ) in matrix form

x 1 x 2a11 a12a21 a22

x 1x 2

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Change of Variables

It is often possible to simplify a QF x Ax by a change of variables x = Sy or y = S 1x , where S is, of course, a nonsingular matrix. We can go fromx to y or from y to x . Substitution of x = Sy into F = x Ax gives F =(Sy ) A(Sy ) = y S ASy = y By , where B = S AS .

Note that if A is a symmetric matrix, B is also symmetric.

The detA is called the discriminant of the QF x Ax . If B = S AS is congruent to A, then the discriminant of the formy By is jB j = jS j jAj jS j = jS j2 jAj;i.e. under a nonsingular change of the variables x = Sy , the discriminantof new QF assumes a magnitude of jS j2 times that of the original form.

The det S is often called the modulus of the transformation x = Sy .DenitionCONGRUENCE: A square matrix B is said to be congruent to the squarematrix A if there exists a nonsingular matrix S such that B = S AS .

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Theorem

If we allow x to vary over all of En , then the set of values taken on by

F = x Ax is called the range of the quadratic form.TheoremUnder a nonsingular transformation of variables, the range of aquadratic form remains unchanged.

Proof.Suppose, we have the form F = x Ax and transform it by the changeof variables x = Sy or y = S 1x to obtain the new form(Sy ) A(Sy ) = y S ASy = y By .Now it is only necessary to note that for any x there is a unique y (and,similarly, for any y there is a unique x ) such that F = x Ax = y By .Hence x Ax and y By must have the same range. In general, thisproperty will not hold if the matrix S is singular. For example, if S = 0,the range of y By contains only a single number 0.


I d i Bili & Q d i F
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Deniteness of QFs - denitions

DenitionsPositive Denite QF : The quadratic form x Ax is said to be positivedenite if it is positive (> 0) for every x except x = 0.Positive Semi-Denite QF : The quadratic form x Ax is said to bepositive semidenite if it is non-negative ( 0) for every x , and there existpoints x 6= 0 for which x Ax = 0.Indenite Forms : A quadratic form x Ax is said to be indenite if theform is positive for some points x and negative for others.

Fact

Negative denite and semidenite forms are dened by interchanging word"negative" and "positive" in the above denitions. If x Ax is positive denite (semidenite), then x ( A)x is negative denite (semidenite).

A symmetric matrix A is often said to be positive denite, positive semidenite, negative denite, etc., if the respective QF: x Ax is positive denite, positive semidenite, negative denite, e tc .[SB ] (Delhi School of Economics) Introductory Math Econ 5th September 2013 6 / 10

I t d ti Bili & Q d ti F
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Another set of necessary and sucient conditions forDeniteness:

A set of necessary and sucient conditions for the form x Ax to bepositive denite is

a11 > 0,a11 a12a21 a22

> 0, 0@

a11 a12 a23a21 a22 a23

a31 a32 a33

1A

> 0, ..., jAj> 0;If these n minors of A are positive, x Ax is positive denite; and x Ax ispositive denite only if these minors are positive.

Fact

(1) x Ax is positive (negative) denite i every eigenvalue of A is positive(negative).(2) x Ax is positive (negative) semidenite i all eigenvalues of A are non-negative (nonpositive), and at least one of the eigenvalues vanishes.(3) x Ax is indenite i A has both positive and negative eigenvalues.


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Examples

(1). F = 3x 21 + 5x 22 , F = 2x 21 + 3x 22 + x 23 , F = x 21 are positive deniteforms in two, three, and one variable, respectively.(2). F = 4x 21 + x 22 4x 1x 2 + 3x 23 = ( 2x 1 x 2)2 + 3x 23 is positivesemidenite since it is never negative and vanishes if x 2 = 2x 1 , x 3 = 0.(3) F = 2x 21 x 22 , F = x 21 x 22 , F = x 21 are negative deniteforms in two, two, and one variable, respectively.(4) F = 4x 21 3x 22 is indenite since it is positive when x 1 = 1, x 2 = 1and negative when x 1 = 0, x 2=1.Note : A positive (negative) denite form remains positive (negative)denite when expressed in terms of a new set of variables provided the transformation of the variables is nonsingular .

Thus if x Ax is positive (negative) denite and S is nonsingular, then(Sy ) A(Sy ) = y S ASy = y By (x = Sy ) is positive (negative) denite.Try to prove this statement given the previous theorem.


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Diagonalisation of QFs

TheoremBy an orthogonal transformation of variables every QF x Ax may be reduced to a diagonal form.

Proof. Given a QF x Ax , consider a nonsingular transformation of variables x = Qy , the matrix Q has as its columns a set of orthonormaleigenvectors of A which span En . The matrix Q is therefore an orthogonalmatrix, and the transformation of variables is called an orthogonaltransformation. In the diagonal form, the coecient of y 2 j is theeigenvalue j of A. In terms of the variables y , the F becomes

y Q AQy = y Dy , and D = jj

j

ij jjis a diagonal matrix whose diagonal elements are the eigenvalues of A.Thus y Dy = n j = 1 j y 2 j . Only the squares of the variables appear;

there are no cross products y i y j . A QF containing only the squares of thevariables is said to be in diagonal form. Furthermore, we say that the

transformation of variables x = Qy has diagonalized the QF x Ax .[SB ] (Delhi School of Economics) Introductory Math Econ 5th September 2013 9 / 10

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Examples

Consider the QF: F = 2x 21 + 2p 2x 1x 2 + x 22 = x Ax the symmetric matrix A is then 2 p 2p 2 1 .To diagonalise the form, we transform x = Qy , where columns of Q are -u 1 = [ 1p 3 , q 23 ]T and u 2 = [q 23 , 1p 3 ]T are orthonormal e.v.s of A.Thus the transformation variables are -x 1 = 1p 3 ( y 1 + p y 2) , y 1 = 1p 3 ( x 1 + p 2x 2)

x 2 = 1p 3 (p 2y 1 + y 2) , y 2 = 1p 3 (p 2x 1 + x 2) .Note that Q 1 = Q ; hence it is very easy to nd the inverse

transformation for an orthogonal transformation of variables. Since theeigenvalues of A are 1 = 0, 2 = 3, the form F becomes F = 3y 22 underthis transformation of variables. The form is therefore positive semidenite.The point y = [2, 0] causes the form to vanish. The x corresponding tothis y is x = [ 2p 3 , 2 2

q23 ], and of course, F vanishes for this value of x .

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002 LS 4 Quadratic Forms