## Quadratic Form in Linear Algebra

An expression

is called quadratic form in the variables x_{1}, x_{2}, x_{3},………….,x_{n} over field F.

Where a_{ij} i=1,2,3,….n, j=1,2,3,….m are elements of F.

If a_{ij} are real then quadratic form is called real quadratic form.

## Examples of Quadratic Form

1-x_{1}^{2}+x_{2}^{2}+ 6 x_{1} x_{2} is a quadratic form in variables x_{1} and x_{2}.

2- x_{1}^{2} + 2x_{2}^{2} + 3x_{3}^{2} + 4x_{1} x_{2}-6 x_{2} x_{3} +8 x_{3} x_{1} is a quadratic form in three variables x_{1}, x_{2} and _{3}.

## Quadratic Form’s Matrix

expression is a quadratic form in variables x_{1}, x_{2}, x_{3},……, x_{n} and X={x_{1}, x_{2}, x_{3},……, x_{n}} then there exists a symmetric matrix A such that

f= X^{T}AX. Here A is said to be matrix of quadratic form.

Examples-

Matrix of the quadratic form in two variables is x_{1}^{2}+x_{2}^{2}+ 6 x_{1} x_{2 } the matrix will be.

Matrix of the quadratic form in three variables is x_{1}^{2} + 2x_{2}^{2} + 3x_{3}^{2} + 4x_{1} x_{2}-6 x_{2} x_{3} +8 x_{3} x_{1 }the matrix will be.

## Quadratic Form’s Classification

expression is a quadratic form in variables x_{1}, x_{2}, x_{3},……, x_{n } the quadratic form f is said to be.

## 1- Positive Definite

If the value of f>0 for all x_{1}, x_{2}, x_{3},……, x_{n}.

And f=0 if x_{1}= x_{2}= x_{3},……,= x_{n} = 0.

## 2- Positive Semi-Definite

If the value of f>0 for all x_{1}, x_{2}, x_{3},……, x_{n. And also f=0 if some vectors from x1, x2, x3,……, xn are not zero.}

## 3- Negative Definite

If the value of f< 0 for all x_{1}, x_{2}, x_{3},……, x_{n. And f=0 if x1= x2= x3,……,= xn=0}

## 4-Negative Semi-Definite

If the value of f<0 for all x_{1}, x_{2}, x_{3},……, x_{n. And also f=0 if some vectors from x1, x2, x3,……, xn are not zero.}

## 5- Indefinite

A quadratic form f is called indefinite if values of f is positive as well as negative for variables x_{1}, x_{2}, x_{3},……, x_{n}.

## Classification of a Quadratic Form Based on Eigen Values

Quadratic form f= X^{T}AX is said to be.

1- Positive definite if all eigen values of matrix A are positive.

2-Negative definite if all eigen values of matrix A are negative.

3- Positive semi-definite if eigen values matrix A are positive and at least one is zero.

4- Negative semi-definite if eigen values matrix A are negative and at least one is zero.

5- Indefinite if eigen values of matrix A are both positive and negative.

Example-

Suppose a quadratic expression is x_{1}^{2} + x_{2}^{2} + 0 x_{3}^{2} then its matrix A and eigen values are 3, 4, 0 which are calculated below.

At least one eigen value is zero and others all eigen values are positive then matrix is positive semi-definite.

## Conclusion-

In this post I have explained about quadratic form which have a lot of application in various domains. Hope you will understand and apply.