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Linear Algebra, Optimisation and Numerical Methods / Linear systems and least squares

Eigenvalues and Covariance Matrices

Portfolio variance w⊤Σw\mathbf{w}^\top\Sigma\mathbf{w} looks entangled when Σ\Sigma has many nonzero covariances. The eigenbasis untangles it: along an eigenvector, Σ\Sigma acts as pure scaling by an eigenvalue, and that eigenvalue is the variance of the corresponding unit eigenportfolio. This note develops eigenvalues for symmetric matrices, states the spectral theorem at working level, and treats a covariance as a symmetric positive semi-definite object whose eigenbasis is the variance basis. Full PSD characterisations follow in Positive Definite Matrices.
Learning objectives. By the end of this note you should be able to:
  • compute eigenpairs of a small symmetric matrix and verify Aq=λqA\mathbf{q}=\lambda\mathbf{q};
  • state the real spectral theorem in matrix form Σ=QΛQ⊤\Sigma=Q\Lambda Q^\top;
  • expand portfolio variance in the eigenbasis and interpret λi\lambda_i as directional variance;
  • explain why a covariance matrix is symmetric and (preview) why its eigenvalues cannot be negative.

In this lesson

  1. Eigenvalues of symmetric matrices
  2. Covariance as a symmetric object
  3. The variance basis

Before you start

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