 # Quadratic Form in Linear Algebra

## Quadratic Form in Linear Algebra

An expression is called quadratic form in the variables x1, x2, x3,………….,xn over field F.

Where aij i=1,2,3,….n, j=1,2,3,….m are elements of F.

If aij are real then quadratic form is called real quadratic form.

1-x12+x22+ 6 x1 x2 is a quadratic form in variables x1 and x2.

2- x12 + 2x22 + 3x32 + 4x1 x2-6 x2 x3 +8 x3 x1 is a quadratic form in three variables x1, x2 and 3. expression is  a quadratic form in variables x1, x2, x3,……, xn and X={x1, x2, x3,……, xn} then there exists a symmetric matrix A such that
f= XTAX. Here A is said to be matrix of quadratic form.

Examples-
Matrix of the quadratic form in two variables is  x12+x22+ 6 x1 x2    the matrix will be. ## 1- Positive Definite

If the value of f>0 for all x1, x2, x3,……, xn.
And f=0 if x1= x2= x3,……,= xn = 0.

## 2- Positive Semi-Definite

If the value of f>0 for all x1, x2, x3,……, xn.
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 x1, x2, x3,……, xn.
And f=0 if x1= x2= x3,……,= xn=0

## 4-Negative Semi-Definite

If the value of f<0 for all x1, x2, x3,……, xn.
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 x1, x2, x3,……, xn.

## Classification of a Quadratic Form Based on Eigen Values

Quadratic form f= XTAX 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 x12 + x22 + 0 x32 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.