A portfolio with many holdings can still be dominated by one common source of risk. Quants use the spectral theorem to turn an asset covariance matrix into uncorrelated risk directions, then measure which of those directions the portfolio actually holds. This is the mathematical foundation for PCA and eigenportfolios: it explains why a complete factor decomposition exists, including when estimated factors have equal variances or the covariance matrix is singular.
The topic develops that guarantee from the real symmetric case to its complex analogue. The lead lesson derives a portfolio-risk decomposition and works through common-movement and spread risk for two assets. The proof lesson explains why extracting one direction does not leave an unsolvable remainder. The final lessons distinguish the assumptions needed for complex data and show why symmetric matrices cannot lack a full eigenbasis. These are the conditions behind the method, rather than extra conditions a risk modeller must hope the data satisfy.
Learning objectives. By the end of this topic you should be able to:
- explain how covariance eigenvectors and eigenvalues translate into factor exposures and portfolio variance;
- state the finite-dimensional real spectral theorem and apply it with repeated or zero eigenvalues;
- follow the proof by invariant-subspace induction and identify why the remaining problem stays self-adjoint;
- distinguish the real symmetric theorem from the complex normal theorem and its Hermitian special case;
- explain why a complete eigenbasis guarantees neither unique factor directions nor an invertible or statistically reliable covariance estimate.
In this lesson
- Contents
- The big picture
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