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Basic Concepts of Quadratic Forms

Definition of a Quadratic Form

The Concept of a Quadratic Form

Let $x_1, x_2, \dots, x_n$ be $n$ variables. A function of the form $Q(x_1, x_2, \dots, x_n) = \sum_{i=1}^n \sum_{j=1}^n a_{ij}x_ix_j$

is called a quadratic form, where $a_{ij}$ are real numbers satisfying $a_{ij} = a_{ji}$ (symmetry).

Examples of Quadratic Forms

Example 1: $Q(x_1, x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ is a quadratic form.

Example 2: $Q(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 3x_3^2 + 2x_1x_2 + 2x_1x_3 + 4x_2x_3$ is a quadratic form.

Matrix Representation of a Quadratic Form

Definition of the Matrix Representation

Matrix Representation of a Quadratic Form

Every quadratic form can be represented as $Q(x) = x^TAx$

where $x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$ and $A$ is a symmetric matrix.

How to Build the Matrix

Steps:

The diagonal entry $a_{ii}$ is the coefficient of $x_i^2$
The off-diagonal entry $a_{ij} = a_{ji}$ is half of the coefficient of $x_ix_j$

Example

Example 3: Write the quadratic form $Q(x_1, x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ in matrix form.

Solution:

Diagonal entries: $a_{11} = 2$ , $a_{22} = 3$
Off-diagonal entries: $a_{12} = a_{21} = \frac{4}{2} = 2$
Matrix representation: $A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$
The quadratic form: $Q(x) = x^TAx = \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$

Basic Properties of Quadratic Forms

Symmetry

Property 1: The matrix representation of a quadratic form is a symmetric matrix.

Proof: Since $a_{ij} = a_{ji}$ , we have $A^T = A$ .

Behaviour Under a Linear Change of Variables

Property 2: If $x = Py$ is a linear change of variables, then $Q(x) = x^TAx = (Py)^TA(Py) = y^T(P^TAP)y$

Classification of Quadratic Forms

Positive definite quadratic form: $Q(x) > 0$ for every nonzero vector $x$
Negative definite quadratic form: $Q(x) < 0$ for every nonzero vector $x$
Positive semidefinite quadratic form: $Q(x) \geq 0$ for every vector $x$
Negative semidefinite quadratic form: $Q(x) \leq 0$ for every vector $x$
Indefinite quadratic form: neither positive definite nor negative definite

Geometric Meaning of Quadratic Forms

Quadratic Forms in Two Dimensions

Geometric meaning: A quadratic form $Q(x_1, x_2) = ax_1^2 + 2bx_1x_2 + cx_2^2$ in two variables describes a conic section in the plane.

Classification:

Positive definite: ellipse
Negative definite: imaginary ellipse
Indefinite: hyperbola
Degenerate: parabola or straight line

Quadratic Forms in Three Dimensions

Geometric meaning: A quadratic form in three variables describes a quadric surface in space.

Classification:

Positive definite: ellipsoid
Negative definite: imaginary ellipsoid
Indefinite: hyperboloid or paraboloid

Reducing a Quadratic Form

Completing the Square

Method: Reduce a quadratic form to a standard form by completing the square.

Example 4: Reduce the quadratic form $Q(x_1, x_2) = x_1^2 + 4x_1x_2 + 4x_2^2$ .

Solution:

$Q(x_1, x_2) = x_1^2 + 4x_1x_2 + 4x_2^2$
$= x_1^2 + 4x_1x_2 + 4x_2^2$
$= (x_1 + 2x_2)^2$
Put $y_1 = x_1 + 2x_2$ , $y_2 = x_2$
Then $Q = y_1^2$

The Orthogonal Transformation Method

Method: Reduce a quadratic form to a standard form by an orthogonal transformation.

Steps:

Write down the matrix $A$ of the quadratic form
Compute the eigenvalues and eigenvectors of $A$
Build the orthogonal matrix $Q$
Make the change of variables $x = Qy$
Obtain the standard form $Q = \lambda_1y_1^2 + \lambda_2y_2^2 + \dots + \lambda_ny_n^2$

Exercises

Exercise 1

Write the matrix representation of the quadratic form $Q(x_1,x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ .

Reference Answer

quadratic formmatrix representationsymmetric matrix

Approach: Build the symmetric matrix of the quadratic form by the standard rule: the diagonal entry collects the coefficient of the squared variable, and each off-diagonal entry collects half of the coefficient of the mixed term.

Steps:

Diagonal entries: $a_{11} = 2$ , $a_{22} = 3$
Off-diagonal entries: $a_{12} = a_{21} = \frac{4}{2} = 2$
Matrix representation: $A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$

Answer: $A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$

Exercise 2

Write the quadratic form $Q(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 3x_3^2 + 2x_1x_2 + 2x_1x_3 + 4x_2x_3$ in matrix form.

Reference Answer

quadratic formmatrix representationcoefficient extraction

Approach: Again build the symmetric matrix directly from the coefficients: diagonal entries come from the squared terms, and every off-diagonal entry is half of the corresponding mixed-term coefficient.

Steps:

Diagonal entries: $a_{11} = 1$ , $a_{22} = 2$ , $a_{33} = 3$
Off-diagonal entries: $a_{12} = a_{21} = \frac{2}{2} = 1$ , $a_{13} = a_{31} = \frac{2}{2} = 1$ , $a_{23} = a_{32} = \frac{4}{2} = 2$
Matrix representation: $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$

Answer: $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$

Exercise 3

Reduce the quadratic form $Q(x_1, x_2) = x_1^2 + 6x_1x_2 + 9x_2^2$ to standard form by completing the square.

Reference Answer

quadratic formcompleting the squarestandard form

Approach: Group the cross term with the squares so that the whole expression becomes a single perfect square, then rename the new combination as a new variable.

Steps:

$Q(x_1, x_2) = x_1^2 + 6x_1x_2 + 9x_2^2$
$= x_1^2 + 6x_1x_2 + 9x_2^2$
$= (x_1 + 3x_2)^2$
Put $y_1 = x_1 + 3x_2$ , $y_2 = x_2$
Then $Q = y_1^2$

Answer: $Q = y_1^2$ , where $y_1 = x_1 + 3x_2$ , $y_2 = x_2$

Exercise 4

Decide whether the quadratic form $Q(x_1, x_2) = x_1^2 + 2x_2^2$ is positive definite.

Reference Answer

quadratic formpositive definitenesssymmetric matrix

Approach: Check the sign of the quadratic form for an arbitrary nonzero vector, using the matrix representation to keep the computation tidy.

Steps:

Matrix representation: $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$
For any nonzero vector $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ , $Q(x) = x_1^2 + 2x_2^2 > 0$
Hence the form is positive definite

Answer:It is a positive definite quadratic form

Exercise 5

Write the matrix representation of the quadratic form $Q(x_1, x_2) = -x_1^2 - 2x_2^2$ and decide its definiteness.

Reference Answer

quadratic formmatrix representationnegative definiteness

Approach: First build the symmetric matrix of the form, then test the sign of the quadratic expression for an arbitrary nonzero vector.

Steps:

Matrix representation: $A = \begin{pmatrix} -1 & 0 \\ 0 & -2 \end{pmatrix}$
For any nonzero vector $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ , $Q(x) = -x_1^2 - 2x_2^2 < 0$
Hence the form is negative definite

Answer: $A = \begin{pmatrix} -1 & 0 \\ 0 & -2 \end{pmatrix}$ , and it is a negative definite quadratic form

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation / Note	Meaning in This Article
$Q(x_1, \dots, x_n)$	Mathematical symbol	Q of x	A quadratic form, a homogeneous quadratic function of the shape $\sum_{i,j} a_{ij}x_ix_j$
$a_{ij}$	Mathematical symbol	a i j	A coefficient of the quadratic form matrix, satisfying the symmetry $a_{ij}=a_{ji}$
$x^TAx$	Mathematical symbol	x transpose A x	The matrix representation of a quadratic form, with $A$ symmetric
$A$	Mathematical symbol	matrix A	The symmetric matrix representing the quadratic form
$A^T = A$	Mathematical symbol	A transpose equals A	The symmetry of the quadratic form matrix
$x = Py$	Mathematical symbol	x equals P y	A linear change of variables, reducing the form to $y^T(P^TAP)y$
$\lambda_i$	Greek letter	Lambda	The coefficient of each squared term in the standard form, that is, an eigenvalue of $A$

Quadratic Forms

Congruence Transformations and Standard Forms

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Basic Concepts of Quadratic Forms

Definition of a Quadratic Form

The Concept of a Quadratic Form

Examples of Quadratic Forms

Matrix Representation of a Quadratic Form

Definition of the Matrix Representation

How to Build the Matrix

Example

Basic Properties of Quadratic Forms

Symmetry

Behaviour Under a Linear Change of Variables

Classification of Quadratic Forms

Geometric Meaning of Quadratic Forms

Quadratic Forms in Two Dimensions

Quadratic Forms in Three Dimensions

Reducing a Quadratic Form

Completing the Square

The Orthogonal Transformation Method

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Summary

Symbols Used in This Article

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