In quantitative finance, eigenvalues and eigenvectors are used in the analysis of financial markets, particularly in the study of portfolio theory and risk management. A practical example is in the application of Principal Component Analysis (PCA) to the covariance matrix of asset returns to identify the principal factors affecting portfolio variance.

In the equation \( A \cdot v = \lambda \cdot v \),
- \( A \) is the matrix.
- \( v \) is the eigenvector.
- \( \lambda \) is the eigenvalue.

This equation tells us that when matrix \( A \) acts on eigenvector \( v \), the output is the same vector \( v \) scaled by the eigenvalue \( \lambda \).

Suppose an investment manager wants to reduce the dimensionality of a dataset of asset returns to identify the underlying factors that explain most of the variance in the returns of a portfolio. Imagine we have a portfolio with two assets, A and B, with the following returns over five periods:

• Asset A: [5%, 10%, 15%, 10%, 5%]
• Asset B: [10%, 20%, 10%, 5%, 0%]

We want to understand the risk structure of this portfolio by performing a Principal Component Analysis (PCA) on the covariance matrix of the returns.

The covariance matrix is given as:

We solve the characteristic equation \( \text{det}(Cov - \lambda I) = 0 \) for eigenvalues \( \lambda \):

Solving this gives eigenvalues:

• \( \lambda_1 = 0.00875 \)
• \( \lambda_2 = 0.0000096875 \)

The corresponding eigenvectors are:

• For \( \lambda_1 = 0.00875 \): \( Eigenvector_1 = [1, 2] \)
• For \( \lambda_2 = 0.0000096875 \): \( Eigenvector_2 = [-2, 1] \)

The eigenvectors represent the directions of variance in the data:

• The first eigenvector (\( [1, 2] \)) shows that when asset A moves by some amount, asset B tends to move by twice that amount in the same direction. This principal component represents a \"market factor\" affecting both assets.
• The second eigenvector (\( [-2, 1] \)) corresponds to minor variance and diversifiable risks, contributing less to the overall risk structure of the portfolio.

The analysis reveals that portfolio risk is predominantly in one dimension, represented by the first eigenvector. Asset B is more volatile and contributes more to portfolio risk, as indicated by the eigenvector \( [1, 2] \).