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OF

MATHEMATICS 101

POSITIVE DEFINITE QUADRATIC FORM OF EQUATIONS

Submitted in the partial fulfillment of the Degree of Bachelor of Technology

(Integrated)

In Mechanical Engineering

SUBMITTED BY:- GUIDED BY:-

Name- NIKHIL LADHA Mr. Jatin Kumar

Regd. No- 11003993

Roll no.- RE4001A23

Section- E4001

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ACKNOWLEDGEMENT

I take this opportunity to present my votes of thanks to all

those guidepost who really acted as lightening pillars to

enlighten our way throughout this project that has led to

successful and satisfactory completion of this study.

We are really grateful to our teacher Mr. Jatin Kumar

for providing us with an opportunity to undertake this project

in this university and providing us with all the facilities. We are

highly thankful to him for his active support, valuable time and

advice, whole-hearted guidance, sincere cooperation and

pains-taking involvement during the study and in completing

the assignment of preparing the said project within the time

stipulated.

Lastly, We are thankful to all those, particularly the various

friends , who have been instrumental in creating proper,

healthy and conductive environment and including new and

fresh innovative ideas for us during the project, their help, it

would have been extremely difficult for us to prepare the

project in a time bound framework.

Name- Nikhil Ladha

Regd. No- 11003993

Roll no. - RE4001A23

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TABLE OF CONTENTS

Quadratic EquationsBinaryQuadratic EquationsPositive Definite FormofquadraticequationsNegative DefiniteQuadratic FormReferences

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QUADRATIC EQUATIONS:-

A quadratic equation is a second-order polynomial equation in a single variable

with . Because it is a second-order polynomial equation, the fundamental theorem of

algebra guarantees that it has two solutions. These solutions may be both real, or

both complex.

The roots can be found by completing the square,

Solving for then gives,

This equation is known as the quadratic formula.

The first known solution of a quadratic equation is the one given in the Berlin papyrus from

the Middle Kingdom (ca. 2160-1700 BC) in Egypt. This problem reduces to solving

An alternate form of the quadratic equation is given by dividing () through by :

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Therefore,

This form is helpful if , where denotes much greater, in which case the usual

form of the quadratic formula can give inaccurate numerical results for one of the roots. Thiscan be avoided by defining

so that and the term under the square root sign always have the same sign. Now, if ,

then

so

Similarly, if , then

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so

Therefore, the roots are always given by and .

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BINARY QUADRATIC EQUATION

A binary quadratic form is a quadratic form in two variables having the form

commonly denoted .

Consider a binary quadratic form with real coefficients , , and , determinant

and . Then is positive definite. An important result states that there exist two

integers and not both 0 such that

for all values of , , and satisfying the above constraint.

FormsofBinaryQuadratic Equations:-

y Positive Definite Form

y Indefinite Definite Form

y NegativedefiniteQuadratic Form

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POSITIVE DEFINITE QUADRATIC FORM OF

EQUATIONS:-

A quadratic form is said to be positive definite if for . A real quadratic

form in variables is positive definite if its canonical form is

A binary quadratic form

of two real variables is positive definite if it is for any , therefore ifand the binary quadratic form discriminant . A binary quadratic form is

positive definite if there exist nonzero and such that

The positive definite quadratic form

is said to be reduced if , , and if or . Under the action ofthe general linear group , i.e., under the set of linear transformations of

coordinates with integer coefficients and determinant , there exists a unique

reduced positive definite binary quadratic form equivalent to any given one.

There exists a one-to-one correspondence between the set of reduced quadratic

forms with fundamental discriminant and the set of classes of fractional ideals of

the unique quadratic field with discriminant . Let be a reduced positive

definite binary quadratic form with fundamental discriminant , and consider the

map which maps the form to the ideal class containing the ideal. Then this map is one-to-one and onto. Thus, the class number of the imaginary

quadratic field is equal to the number of reduced binary quadratic forms of

discriminant , which can be easily computed by systematically constructing all

binary quadratic forms of discriminant by looping over the coefficients and . The

third coefficient is then determined by , , and .

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A quadratic form is positive definite if every eigenvalue of is positive. A

quadratic form with a Hermitian matrix is positive definite if all the

principal minors in the top-left corner of are positive, in other words

(1) Ifq(~x) > 0 forall nonzero ~x in reaction, we say A is positi e definite.

(2)If

q(~x) _ 0for

all no

nzero

~x inreacti

on, we say A is p

ositi e semi de

finite.(3) Ifq(~x) takes positi e as well as negati e val es, we say A is indefinite.

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Example 1.

The matrix is positive definite. Fora vectorwith entries

the quadratic form is

when the entries z0, z1 are real and atleastone ofthem nonzero, this is positive.

The matrix is not positive definite. When the

quadratic form atzis then

Another example ofpositive definite matrix is given by

Itis positive definite since forany non-zero vector , we have

Fora huge class ofexamples, considerthatin statistics positive definite matrices

appearas covariance matrices. In fact all positive definite matrices are a covariance

matrix forsome probability distribution.

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E l 2.

Su os q(X)=(X^T)AX wh r A is sy tric. Prov th t if ll ig nv lu s of A rositiv , th n q is ositiv d finit (i. . q(X)>0 for ll X not =0).

Proof:

Since A is symmetric, by principal axis theorem, there exists an orthogonal matrix P such that(P^T)AP=diag{c1,c2,...,cn} is diagonal, where c1,...,cn are eigenvalues of A.

Suppose ci>0 for all i=1,...,n

For any X not =0, X=PY

This implies that Y not =0

=> q=(X^T)AX=[(PY)^T]A(PY)=c1(y1)^2+...+cn(yn)^2 > 0 for all X not =0 since ci>0 and

Y not =0

E l 3.

Consider m x n matrix A. Show that the function q(~x) = jjA~xjj2 is a quadratic form, find itsmatrix and determine its definite-ness.

Solution: q(~x) = (A~x) _ (A~x) = (A~x)T (A~x) = ~xTATA~x = ~x _ (ATA~x).

This shows that q is a quadratic form, with symmetric matrix ATA.Since q(~x) = jjA~xjj2 _ 0 for all vectors ~x in Rn, this quadratic form is positive semi

definite.

Note that q(~x) = 0 i_ ~x is in the kernel of A. Therefore, the quadratic form is positivedefinite iff ker(A) = {0~}.

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NEGATIVE DEFINITE QUADRATIC FORM:-

If x'Ax < 0 forallx 0, the quadratic form is not positive definite.

Example 1:

x12

+ 2x1x2 + x22

may be expressed as (x1 + x2)2, whichis nonnegative forall (x1, x2).

Thus this quadratic form is positive semi definite. Itis not positive definite because

(x1 + x2)2

= 0 for(x1, x2) = (1,1) (forexample).

Example 2:

Let,

The leading principal minors ofA are D1 = 3 < 0, D2 =

(3)(3)(2)(2) = 5 > 0, and |A| = 25 < 0. Thus Ais negative definite.

Two variables

We can easily derive conditions forthe definiteness of any quadratic form in two variables.

To make the argument more readable, I change the notation slightly, usingx and yfor the

variables, ratherthan x1 and x2. Considerthe quadratic form

Q(x, y) = ax2 + 2bxy + cy2.

Ifa = 0 then Q(1,0) = 0, soQis neitherpositive nornegative definite. So assume thata 0.

Given a 0, we have

Q(x, y) = a[(x + (b/a)y)2

+ (c/a (b/a)2)y

2].

Both squares are always nonnegative, and at leastone ofthem is positive unless(x, y) = (0,

0). Thus ifa > 0 and c/a (b/a)2

> 0 then Q(x, y) is positive definite. Given a > 0, the second

condition is ac > b

2

.Th

us we conclude that ifa > 0 and ac > b

2

then Q(x, y) is positive

definite.

Now, we have Q(1, 0) = a and Q(b/a, 1) = (ac b2)/a. Thus, ifQ(x, y) is positive definite

then a > 0 and ac > b2.

We conclude thatQ(x, y) is positive definite if and only ifa > 0 andac > b2.

A similarargument shows thatQ(x, y) is negative definite if and only ifa < 0 andac > b2.

A =

3 2 0

2 3 0

0 0 5

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Note that ifa > 0 and ac > b2then because b

2 0 for allb, we can conclude thatc > 0.

Similarly, ifa < 0 and ac > b2then c < 0. Thus, to determine whether a quadratic form is

positive or negative definite we need to lookonly atthe signs ofa and ofac b2, but ifthe

conditions for positive definiteness are satisfied then it must in fact also be true thatc > 0,

and ifthe conditions fornegative definitely are satisfied then we must alsohavec < 0.

Notice thatac b2is the determinantofthe matrix thatrepresents the quadratic form, namely

A =

a b

b c

Thus we can rewrite the results as follows: the two variable quadratic formQ(x, y) = ax2

+

2bxy + cy2is

y positive definite ifand only ifa > 0 and |A| > 0 (in which case c > 0)

y negative definite ifand only ifa < 0 and |A| > 0 (in which case c < 0)

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REFERENCES

www.google.com Universal search website

www.wikipedia.com Educational search website

www.mathworld.com Mathematics search website