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Linear Algebra, Optimisation and Numerical Methods / Spectral and factor methods

Positive Definite Matrices

Portfolio variance w⊤Σw\mathbf{w}^\top\Sigma\mathbf{w} must be non-negative for every book w\mathbf{w}. That single inequality is the definition of positive semi-definiteness, and it is equivalent to "all eigenvalues ≥0\ge 0", to "a Cholesky factor exists" (in the definite case), and to "the matrix is a Gram matrix S⊤SS^\top S". This note states those equivalences at working level, connects them to quadratic forms, and reads a negative eigenvalue of a supposed covariance as a desk failure mode — not a market fact.
Learning objectives. By the end of this note you should be able to:
  • state equivalent characterisations of PSD / PD for symmetric matrices;
  • test a small matrix via eigenvalues or the quadratic form;
  • state the existence (and uniqueness with positive diagonal) of a Cholesky factor for PD matrices;
  • explain why a covariance must be PSD and what λ<0\lambda<0 implies for reported portfolio variance.

In this lesson

  1. Definitions and equivalences
  2. Quadratic forms and portfolio variance
  3. Cholesky (existence statement)

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

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