Eigenvalues and Eigenvectors

Motivation: why this matters in quant finance

Risk systems need to know which portfolio directions a covariance matrix stretches most. Eigenvectors are those directions; eigenvalues are the stretch factors. PCA, stress testing, and term-structure factor models all begin with this question.

The informal idea

An eigenvector is a direction preserved by a matrix. The matrix may stretch or reverse the vector, but it does not rotate it into a different line.

Formal definitions

For a square matrix $A$, a non-zero vector $\mathbf{v}$ is an eigenvector with eigenvalue $\lambda$ when $A\mathbf{v}=\lambda\mathbf{v}$.

Key properties

Eigenvectors expose natural coordinates

In an eigenbasis, a linear map acts by scaling coordinates.

Symmetric matrices have orthogonal eigenvectors

Covariance matrices can be diagonalised with orthogonal risk directions.

Eigenvalues of covariance matrices are variances

Large eigenvalues identify dominant modes of portfolio movement.

Worked example

For $A=\begin{bmatrix}2&0\\0&5\end{bmatrix}$, the coordinate vectors are eigenvectors. The first direction is scaled by $2$ and the second by $5$.

Common confusions and pitfalls

"Eigenvectors are components." They are directions, not individual assets.

"The largest eigenvalue is always the best factor." It is the largest variance direction, not necessarily an economic factor.

"Non-square matrices have eigenvalues." Rectangular matrices use singular values.

Where this goes next

• Positive Definite Matrices: connects positive eigenvalues with valid covariance matrices.
• Matrix Factorisations: uses eigendecomposition and SVD.
• Principal Component Analysis for Risk Factors: applies eigenvectors to risk factors.

References

• Lang, S. (1986). Introduction to Linear Algebra (2nd ed.). Springer. Ch. II, Ch. V-VIII, and the eigenvalue chapter as relevant.