This is a beta course, so its structure, chapters, and examples may continue to change.

Positive Definiteness

Positive Definite Quadratic Forms

Definition of a Positive Definite Quadratic Form

Let $Q(x)$ be a quadratic form in $n$ variables. If for every nonzero vector $x$ we have $Q(x) > 0$

then $Q(x)$ is called a positive definite quadratic form.

Definition of a Positive Definite Matrix

Let $A$ be a real symmetric matrix of order $n$ . If for every nonzero vector $x$ we have $x^TAx > 0$

then $A$ is called a positive definite matrix.

Examples of Positive Definite Quadratic Forms

Example 1: $Q(x_1, x_2) = x_1^2 + 2x_2^2$ is a positive definite quadratic form.

Example 2: $Q(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2$ is a positive definite quadratic form.

Criteria for Positive Definiteness

The Eigenvalue Criterion

Eigenvalue Criterion

A real symmetric matrix $A$ is positive definite if and only if all of its eigenvalues are positive.

Proof

Proof:

Sufficiency: Suppose all eigenvalues of $A$ are positive. Then by orthogonal diagonalisation $A = QDQ^T$ , where the diagonal entries of $D$ are positive. For any nonzero vector $x$ , $x^TAx = x^TQDQ^Tx = y^TDy > 0$ , where $y = Q^Tx$ .
Necessity: Suppose $A$ is positive definite, $\lambda$ is an eigenvalue of $A$ , and $\vec{x}$ is the corresponding eigenvector. Then $0 < \vec{x}^TA\vec{x} = \lambda\vec{x}^T\vec{x}$ , so $\lambda > 0$ .

The Leading Principal Minor Criterion

Leading Principal Minor Criterion

A real symmetric matrix $A$ is positive definite if and only if all of its leading principal minors are positive.

Let $A = (a_{ij})$ be a matrix of order $n$ . Then

$A_1 = a_{11}$
$A_2 = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}$
$\dots$
$A_n = |A|$

are called the leading principal minors of $A$ .

Comparing the Two Criteria

Criterion	Advantage	Drawback
Eigenvalue criterion	Theoretically complete	Computationally involved
Leading principal minor criterion	Easy to compute	Only applies to real symmetric matrices

Negative Definiteness and Semidefiniteness

Negative Definite Quadratic Forms

Let $Q(x)$ be a quadratic form in $n$ variables. If for every nonzero vector $x$ we have $Q(x) < 0$

then $Q(x)$ is called a negative definite quadratic form.

Positive Semidefinite Quadratic Forms

Let $Q(x)$ be a quadratic form in $n$ variables. If for every vector $x$ we have $Q(x) \geq 0$

then $Q(x)$ is called a positive semidefinite quadratic form.

Negative Semidefinite Quadratic Forms

Let $Q(x)$ be a quadratic form in $n$ variables. If for every vector $x$ we have $Q(x) \leq 0$

then $Q(x)$ is called a negative semidefinite quadratic form.

Indefinite Quadratic Forms

A quadratic form that is neither positive definite nor negative definite is called an indefinite quadratic form.

Applications of Positive Definiteness

Applications in Optimisation

Application 1: optimising a quadratic function Let $f(x) = \frac{1}{2}x^TAx + b^Tx + c$ , where $A$ is a positive definite matrix. Then $f(x)$ has a unique minimiser.

Application 2: Newton’s method In Newton’s method, the algorithm converges when the Hessian matrix is positive definite.

Applications in Geometry

Application 3: ellipsoids A positive definite quadratic form $Q(x) = x^TAx$ describes an ellipsoid.

Application 4: distance metrics A positive definite matrix can be used to define a distance metric.

Applications in Numerical Analysis

Application 5: matrix factorisation A positive definite matrix admits a Cholesky decomposition: $A = LL^T$ , where $L$ is a lower triangular matrix.

Properties of Positive Definite Matrices

Basic Properties

Property 1: The determinant of a positive definite matrix is positive.

Property 2: The main diagonal entries of a positive definite matrix are all positive.

Property 3: The inverse of a positive definite matrix is also positive definite.

Property 4: The sum of two positive definite matrices is also positive definite.

Proofs

Proof of Property 1: Let $A$ be a positive definite matrix. All eigenvalues of $A$ are positive, so $|A| = \prod_{i=1}^n \lambda_i > 0$ .

Proof of Property 2: Let $A$ be a positive definite matrix and let $e_i$ be the $i$ -th standard basis vector. Then $0 < e_i^TAe_i = a_{ii}$ .

Exercises

Exercise 1

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

Reference Answer

positive definitenessquadratic formsymmetric matrix

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

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 2

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

Reference Answer

positive definitenesseigenvalue criterionleading principal minors

Approach: Use the eigenvalue criterion and the leading principal minor criterion; either one settles the question, and computing both is a useful consistency check.

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 3

Decide the positive definiteness of the quadratic form $Q(x_1, x_2) = -x_1^2 - 2x_2^2$ .

Reference Answer

positive definitenessnegative definitenessquadratic form

Approach: Evaluate the sign of the form on an arbitrary nonzero vector; a sign that is always negative settles the classification at once.

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:It is a negative definite quadratic form

Exercise 4

Decide the positive definiteness of the matrix $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ .

Reference Answer

positive definitenesseigenvalue criterionpositive semidefinite

Approach: Compute the eigenvalues. A zero eigenvalue rules out positive definiteness, while the absence of negative ones still leaves a semidefinite conclusion.

Steps:

Characteristic equation: $|A - \lambda I| = \lambda^2 - 2\lambda = 0$
Eigenvalues: $\lambda_1 = 0$ , $\lambda_2 = 2$
There is a zero eigenvalue, so the matrix is not positive definite
But all eigenvalues are nonnegative, so it is positive semidefinite

Answer:It is a positive semidefinite matrix

Exercise 5

Prove that the determinant of a positive definite matrix is positive.

Reference Answer

positive definitenessdeterminanteigenvalues

Approach: Use the eigenvalue property of positive definite matrices together with the fact that the determinant of a matrix equals the product of its eigenvalues.

Steps:

Let $A$ be a positive definite matrix
All eigenvalues of a positive definite matrix are positive
The determinant is the product of the eigenvalues: $|A| = \prod_{i=1}^n \lambda_i$
All eigenvalues are positive, so their product is positive as well
Therefore $|A| > 0$

Answer:The proof is complete

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation / Note	Meaning in This Article
$x^TAx$	Mathematical symbol	x transpose A x	A quadratic form; positive definite means $x^TAx>0$ for every nonzero $x$
$Q(x) > 0$	Mathematical symbol	Q of x greater than zero	The definition of a positive definite quadratic form
$A = QDQ^T$	Mathematical symbol	A equals Q D Q transpose	The orthogonal diagonalisation of a real symmetric matrix, with the eigenvalues on the diagonal of $D$
$\lambda$	Greek letter	Lambda	An eigenvalue of $A$ ; $A$ is positive definite if and only if every $\lambda>0$
$A_i$	Mathematical symbol	A i	A leading principal minor of $A$ ; $A$ is positive definite if and only if every $A_i>0$
$e_i$	Mathematical symbol	e i	The $i$ -th standard basis vector, used to prove $a_{ii}>0$
$A = LL^T$	Mathematical symbol	A equals L L transpose	The Cholesky decomposition of a positive definite matrix, with $L$ lower triangular

Congruence Transformations and Standard Forms

Comprehensive Exercises on Quadratic Forms

Found an issue with this content?

Tell us what could be improved.

Unlock Full Course ·

Your membership covers this course — no need to buy it again.

Login Required to Purchase

Payment Failed

Payment Successful!

Price History

Positive Definiteness

Positive Definite Quadratic Forms

Definition of a Positive Definite Quadratic Form

Definition of a Positive Definite Matrix

Examples of Positive Definite Quadratic Forms

Criteria for Positive Definiteness

The Eigenvalue Criterion

The Leading Principal Minor Criterion

Comparing the Two Criteria

Negative Definiteness and Semidefiniteness

Negative Definite Quadratic Forms

Positive Semidefinite Quadratic Forms

Negative Semidefinite Quadratic Forms

Indefinite Quadratic Forms

Applications of Positive Definiteness

Applications in Optimisation

Applications in Geometry

Applications in Numerical Analysis

Properties of Positive Definite Matrices

Basic Properties

Proofs

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Summary

Symbols Used in This Article

Send feedback