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Eigenvalues and Eigenvectors

Eigenvalues are where linear algebra meets risk. The eigenvectors of a covariance matrix are portfolios whose returns are uncorrelated with each other, and the eigenvalues are their variances. Interviewers use this to ask why an optimiser loves a spread between two correlated assets, what a negative eigenvalue means, and how principal component analysis works. This lesson builds the tools; the next lesson applies them.

The definition

A vector vv is an eigenvector of a square matrix AA when multiplying by AA only stretches it:

Av=λvA v = \lambda v

The number λ\lambda is the eigenvalue. Most vectors change direction when multiplied by a matrix. Eigenvectors are the special directions that do not.

To find the eigenvalues, solve det⁡(A−λI)=0\det(A - \lambda I) = 0. For a 2×22 \times 2 matrix this is the quadratic λ2−tr⁡(A) λ+det⁡(A)=0\lambda^2 - \operatorname{tr}(A)\,\lambda + \det(A) = 0, where the trace tr⁡(A)\operatorname{tr}(A) is the sum of the diagonal.

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