Matrix of a Quadratic Form in 3 Variables

Quadratic Form in Linear Algebra

Quadratic Form in Linear Algebra

An expression

Quadratic Form 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.

Examples of 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.

Quadratic Form’s Matrix

Quadratic Form Expression

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.

Matrix of a Quadratic Form in 2 Variables
Matrix of the quadratic form in three  variables is x12 + 2x22 + 3x32 + 4x1 x2-6 x2 x3 +8 x3 x1 the matrix will be.

Matrix of a Quadratic Form in 3 Variables

Quadratic Form’s Classification

Quadratic Form Expression
expression is  a quadratic form in variables x1, x2, x3,……, x     the quadratic form f is said to 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.

Classification of Quadratic Forms Based On Eigen Values

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.

 

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