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Congruence Transformations and Standard Forms

Congruence Transformations

Definition of a Congruence Transformation

Let $A$ and $B$ be real symmetric matrices of order $n$ . If there is an invertible matrix $P$ such that $B = P^TAP$

then $A$ and $B$ are called congruent, written $A \cong B$ .

Properties of Congruence Transformations

Property 1: Congruence is an equivalence relation (reflexive, symmetric, transitive).

Property 2: A congruence transformation preserves the symmetry of a matrix.

Property 3: A congruence transformation preserves the rank of a matrix.

Property 4: A congruence transformation preserves the positive definiteness of a matrix.

Geometric Meaning of Congruence Transformations

Geometric meaning: A congruence transformation corresponds to a linear change of coordinate system.

Example: Let $Q(x) = x^TAx$ be a quadratic form and make the change of variables $x = Py$ . Then $Q(x) = x^TAx = (Py)^TA(Py) = y^T(P^TAP)y = y^TBy$

where $B = P^TAP$ is a congruence transformation.

Standard Forms

Definition of a Standard Form

If the quadratic form $Q(x)$ can be reduced by a congruence transformation to $Q(x) = \lambda_1y_1^2 + \lambda_2y_2^2 + \dots + \lambda_ny_n^2$

then this form is called a standard form.

How to Obtain a Standard Form

Method 1: the orthogonal transformation method

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 $Q = \lambda_1y_1^2 + \lambda_2y_2^2 + \dots + \lambda_ny_n^2$

Method 2: completing the square

Steps:

Eliminate the cross terms step by step by completing the square
Obtain the standard form

Example

Example 1: Reduce the quadratic form $Q(x_1, x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ to standard form.

Solution (orthogonal transformation method):

Matrix representation: $A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$
Characteristic equation: $|A - \lambda I| = \lambda^2 - 5\lambda + 2 = 0$
Eigenvalues: $\lambda_1 = 1$ , $\lambda_2 = 4$
Eigenvectors:
- $\lambda_1 = 1$ : $\vec{x}_1 = (1, -1)$
- $\lambda_2 = 4$ : $\vec{x}_2 = (1, 1)$
Normalisation:
- $\vec{q}_1 = \frac{1}{\sqrt{2}}(1, -1)$
- $\vec{q}_2 = \frac{1}{\sqrt{2}}(1, 1)$
Orthogonal matrix: $Q = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$
Standard form: $Q = y_1^2 + 4y_2^2$

The Law of Inertia

Statement of the Law of Inertia

Theorem (law of inertia): Let $A$ be a real symmetric matrix of order $n$ . The positive and negative indices of inertia of $A$ are unchanged under congruence transformations.

Definition of the Indices of Inertia

Positive index of inertia: the number of positive coefficients in the standard form
Negative index of inertia: the number of negative coefficients in the standard form
Zero index of inertia: the number of zero coefficients in the standard form

Proof of the Law of Inertia

Proof strategy:

Suppose $A$ is reduced to standard forms by two different congruence transformations
Prove that the positive and negative indices of the two standard forms coincide
Use the geometric properties of quadratic forms to do so

Applications of the Law of Inertia

Application 1: deciding the positive definiteness of a quadratic form

Positive definite: the positive index of inertia equals $n$ and the negative index of inertia equals $0$
Negative definite: the positive index of inertia equals $0$ and the negative index of inertia equals $n$
Positive semidefinite: the negative index of inertia equals $0$
Negative semidefinite: the positive index of inertia equals $0$

Application 2: deciding the positive definiteness of a matrix

Positive definite: all eigenvalues are positive
Negative definite: all eigenvalues are negative
Positive semidefinite: all eigenvalues are nonnegative
Negative semidefinite: all eigenvalues are nonpositive

Completing the Square

The Basic Idea

Idea: Eliminate the cross terms step by step by completing the square, until the standard form is reached.

Steps

Steps:

Choose one variable and complete the square for its squared term together with the cross terms
Introduce a new variable to replace the completed expression
Repeat the process until all cross terms are eliminated
Obtain the standard form

Example

Example 2: Reduce the quadratic form $Q(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 3x_3^2 + 2x_1x_2 + 2x_1x_3 + 4x_2x_3$ to standard form by completing the square.

Solution:

$Q = x_1^2 + 2x_1x_2 + 2x_1x_3 + 2x_2^2 + 4x_2x_3 + 3x_3^2$
$= (x_1 + x_2 + x_3)^2 - x_2^2 - x_3^2 - 2x_2x_3 + 2x_2^2 + 4x_2x_3 + 3x_3^2$
$= (x_1 + x_2 + x_3)^2 + x_2^2 + 2x_2x_3 + 2x_3^2$
$= (x_1 + x_2 + x_3)^2 + (x_2 + x_3)^2 + x_3^2$
Put $y_1 = x_1 + x_2 + x_3$ , $y_2 = x_2 + x_3$ , $y_3 = x_3$
Then $Q = y_1^2 + y_2^2 + y_3^2$

Exercises

Exercise 1

Reduce the quadratic form $Q(x_1,x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ to standard form by an orthogonal transformation.

Reference Answer

congruence transformationorthogonal transformationstandard formeigenvalues

Approach: Use the orthogonal transformation method: diagonalise the symmetric matrix of the quadratic form, normalise the eigenvectors to get an orthogonal matrix, and read the standard form off the eigenvalues.

Steps:

Matrix representation: $A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$
Characteristic equation: $|A - \lambda I| = \lambda^2 - 5\lambda + 2 = 0$
Eigenvalues: $\lambda_1 = 1$ , $\lambda_2 = 4$
Eigenvectors:
- $\lambda_1 = 1$ : $\vec{x}_1 = (1, -1)$
- $\lambda_2 = 4$ : $\vec{x}_2 = (1, 1)$
Normalisation:
- $\vec{q}_1 = \frac{1}{\sqrt{2}}(1, -1)$
- $\vec{q}_2 = \frac{1}{\sqrt{2}}(1, 1)$
Orthogonal matrix: $Q = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$
Standard form: $Q = y_1^2 + 4y_2^2$

Answer: $Q = y_1^2 + 4y_2^2$

Exercise 2

Reduce the quadratic form $Q(x_1, x_2) = x_1^2 + 4x_1x_2 + 4x_2^2$ to standard form by completing the square.

Reference Answer

completing the squarestandard formquadratic form

Approach: Complete the square so that the cross term disappears into a single squared expression, then rename that expression as a new variable.

Steps:

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

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

Exercise 3

State the law of inertia.

Reference Answer

law of inertiaindices of inertiacongruence transformation

Approach: Recall the statement of the law of inertia, which links the symmetry of a matrix with the invariance of its signature under congruence.

Steps:

The law of inertia: let $A$ be a real symmetric matrix of order $n$ . Then the positive index of inertia and the negative index of inertia of $A$ are unchanged under congruence transformations.

Answer:Congruence transformations do not change the positive or negative indices of inertia

Exercise 4

Decide whether the matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ is positive definite.

Reference Answer

positive definitenesseigenvalue criterionleading principal minors

Approach: Apply the eigenvalue criterion and, as a cross-check, the criterion based on the leading principal minors; either test decides the question on its own.

Steps:

Method one: compute the eigenvalues
- Characteristic equation: $|A - \lambda I| = \lambda^2 - 4\lambda + 3 = 0$
- Eigenvalues: $\lambda_1 = 1$ , $\lambda_2 = 3$ , both positive
Method two: check the leading principal minors
- $|A_1| = 2 > 0$
- $|A_2| = 3 > 0$

Answer:It is a positive definite matrix

Exercise 5

Reduce the quadratic form $Q(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 3x_3^2 + 2x_1x_2 + 2x_1x_3 + 4x_2x_3$ to standard form by the orthogonal transformation method.

Reference Answer

orthogonal transformationstandard formeigenvalues and eigenvectors

Approach: Diagonalise the symmetric matrix of the form, normalise the eigenvectors so that they form an orthogonal basis, and then read the standard form off the eigenvalues.

Steps:

Matrix representation: $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$
Characteristic equation: $|A - \lambda I| = -\lambda^3 + 6\lambda^2 - 11\lambda + 6 = 0$
Eigenvalues: $\lambda_1 = 1$ , $\lambda_2 = 2$ , $\lambda_3 = 3$
Eigenvectors:
- $\lambda_1 = 1$ : $\vec{x}_1 = (1, -1, 0)$
- $\lambda_2 = 2$ : $\vec{x}_2 = (1, 0, -1)$
- $\lambda_3 = 3$ : $\vec{x}_3 = (1, 1, 1)$
Normalisation:
- $\vec{q}_1 = \frac{1}{\sqrt{2}}(1, -1, 0)$
- $\vec{q}_2 = \frac{1}{\sqrt{2}}(1, 0, -1)$
- $\vec{q}_3 = \frac{1}{\sqrt{3}}(1, 1, 1)$
Standard form: $Q = y_1^2 + 2y_2^2 + 3y_3^2$

Answer: $Q = y_1^2 + 2y_2^2 + 3y_3^2$

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation / Note	Meaning in This Article
$B = P^TAP$	Mathematical symbol	B equals P transpose A P	A congruence transformation: an invertible matrix $P$ turns $A$ into $B$
$A \cong B$	Mathematical symbol	A congruent B	The notation for the congruence of $A$ and $B$
$P$	Mathematical symbol	matrix P	The invertible matrix used in the congruence transformation
$\lvert A - \lambda I\rvert = 0$	Mathematical symbol	determinant of A minus lambda I	The characteristic equation used to find the eigenvalues
$\lambda_1, \lambda_2$	Greek letter	Lambda	The eigenvalues of the matrix, and the coefficients of the squared terms in the standard form
$\vec{q}_1, \vec{q}_2$	Mathematical symbol	q one, q two	The normalised eigenvectors, forming the columns of the orthogonal matrix $Q$
$\sqrt{2}$	Mathematical symbol	square root of two	The normalisation factor $\frac{1}{\sqrt{2}}$ used when scaling a vector to unit length

Basic Concepts of Quadratic Forms

Positive Definiteness

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Congruence Transformations and Standard Forms