Positive Definite Matrices

Motivation: why this matters in quant finance

A covariance matrix must never assign negative variance to a portfolio. Positive definiteness is the matrix condition that enforces this and supports stable quadratic optimisation.

The informal idea

A symmetric matrix is positive definite when every non-zero direction has positive quadratic form. Geometrically, it bends space like a bowl rather than a saddle.

Formal definitions

A symmetric matrix $A$ is positive definite if $\mathbf{x}^\top A\mathbf{x}>0$ for every non-zero $\mathbf{x}$. It is positive semidefinite if the inequality is $\ge 0$.

Key properties

Quadratic risk is non-negative

Portfolio variance $\mathbf{w}^\top\Sigma\mathbf{w}$ is valid only when $\Sigma$ is positive semidefinite.

Eigenvalues test definiteness

A symmetric matrix is positive definite exactly when all eigenvalues are positive.

Cholesky requires positive definiteness

Strictly positive definite covariance matrices admit $\Sigma=LL^\top$.

Worked example

For $A=\begin{bmatrix}2&0\\0&3\end{bmatrix}$, $\mathbf{x}^\top A\mathbf{x}=2x_1^2+3x_2^2>0$ unless $\mathbf{x}=\mathbf{0}$.

Common confusions and pitfalls

"Positive entries imply positive definite." Entry signs are not enough.

"A sample covariance is always invertible." It may be singular when assets outnumber observations.

"Semidefinite and definite are interchangeable." Optimisation and inversion often require strict definiteness.

Where this goes next

• Matrix Factorisations: uses Cholesky and spectral decompositions.
• Mean-Variance Optimisation: depends on positive covariance risk.
• Convex Optimization: uses positive semidefinite Hessians.

References

• Lang, S. (1986). Introduction to Linear Algebra (2nd ed.). Springer. Ch. II, Ch. V-VIII, and the eigenvalue chapter as relevant.