Positive Definite Matrices

A matrix is positive definite when every quadratic form it induces is positive — the algebraic signature of a valid variance or a strict minimum.

1. The idea

A symmetric matrix $A$ is positive definite (PD) if $x^{T} A x > 0$ for all nonzero $x$ , and positive semi-definite (PSD) if $x^{T} A x \geq 0$ . Geometrically, PD matrices describe quadratic 'bowls' with a unique minimum.

Equivalent tests: all eigenvalues are positive (PD) or non-negative (PSD); all leading principal minors are positive (PD, Sylvester's criterion); or $A = B^{T} B$ for some matrix $B$ with full column rank (PD).

Covariance matrices are always PSD by construction, since $x^{T} Σ x = Var (x^{T} R) \geq 0$ for any return vector $R$ ; they fail to be strictly PD only when some linear combination of assets has zero variance.

2. Why it matters

Portfolio optimization, PCA and mean-variance analysis all require checking positive definiteness to ensure a covariance matrix is valid and a quadratic optimization has a unique solution.

3. Core mathematics

Definition

x^{T} A x > 0 \forall x \neq = 0 ⟺ A PD

Eigenvalue test

A PD ⟺ λ_{i} > 0 \forall i

Cholesky form

A = L L^{T}

exists iff $A$ is PD

Covariance is PSD

x^{T} Σ x = Var (x^{T} R) \geq 0

4. How to recognize it

Think of this when…

Checking whether a proposed covariance matrix is valid
Confirming a quadratic optimization problem has a unique minimum
Any mention of 'Cholesky decomposition' or 'quadratic form'

5. Worked example

Worked example

Is $A = (2112)$ positive definite?

1
Check leading principal minors: top-left entry.
$2 > 0$
2
Check the full determinant.
$det (A) = 4 - 1 = 3 > 0$
3
Both minors positive, so by Sylvester's criterion, $A$ is PD.
Answer: Yes, A is positive definite (eigenvalues 1 and 3, both positive).

6. Common mistakes

Confusing positive semi-definite (allows zero) with strictly positive definite
Checking only that diagonal entries are positive, ignoring off-diagonal structure
Assuming any symmetric matrix with positive determinant is PD (false for 3x3 and larger without checking all minors)

7. Quick check

A covariance matrix is always:

If all eigenvalues of a symmetric matrix are strictly positive, the matrix is:

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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