Exercise 3 of the differential-equations note found a matrix with complex eigenvalues; the defective-matrix example found one that couldn't be diagonalised at all. This note proves that symmetric matrices are immune to both failure modes: their eigenvalues are always real, and eigenvectors for distinct eigenvalues are always orthogonal. Since covariance and correlation matrices are always symmetric, this single fact is the reason risk decomposition in finance never has to worry about complex or non-diagonalisable eigenstructure.
Learning objectives. By the end of this note you should be able to:
- prove that a symmetric matrix's eigenvalues are always real;
- prove that eigenvectors of a symmetric matrix for distinct eigenvalues are orthogonal;
- state the spectral theorem (preview) and identify what it guarantees beyond this note's two facts;
- explain why this matters specifically for covariance matrices.
In this lesson
- Real eigenvalues
- Orthogonal eigenvectors
- The spectral theorem, previewed
Before you start
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