Positive Definite Matrices
xᵀAx > 0 for all x ≠ 0.
Key idea
xᵀAx > 0 for all x ≠ 0.
In depth
A symmetric A is positive definite if xᵀAx > 0 for all x ≠ 0, equivalently all eigenvalues are positive, or all leading minors are positive. Such matrices define valid inner products and bowl-shaped energy functions.
Worked example
[2 −1; −1 2] has eigenvalues 1 and 3, so is positive definite.
Real-world uses
- Checking that an energy function has a minimum, as in optimisation and statistics
Common mistake
Checking only the diagonal entries, which is not sufficient.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Positive Definite Matrices?
xᵀAx > 0 for all x ≠ 0. It is a topic at the expert level in Linear Algebra.
Where is Positive Definite Matrices used in real life?
Real-world examples: Checking that an energy function has a minimum, as in optimisation and statistics.
What should I learn before Positive Definite Matrices?
Make sure you are comfortable with Spectral Theorem, the previous topic in Linear Algebra.
What is a common mistake with Positive Definite Matrices?
Checking only the diagonal entries, which is not sufficient.
How can I practise Positive Definite Matrices?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Positive Definite Matrices?
Move on to Quadratic Forms next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.