Brownian Bridge™

Setup

Where eigendecomposition appears in quant finance

Three workflows on a quant desk depend directly on eigendecomposition:

Principal component analysis (PCA) of yield curves. The covariance matrix of daily rate changes is decomposed into eigenvectors: the first three principal components (level, slope, curvature) typically explain over 95% of the variance. Traders hedge using these components rather than individual maturities.
Risk factor decomposition. A covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ for $n$ equity returns is decomposed as $\Sigma = Q\Lambda Q^\top$ . The eigenvalues $\lambda_i$ are the variances of uncorrelated risk factors; the eigenvectors $q_i$ are the factor loadings. The smallest eigenvalues flag near-arbitrage portfolios; the largest flag dominant market risk.
Stability of calibration. The condition number of a matrix (treated in Module 4) is the ratio of its largest to smallest eigenvalue. A large condition number signals that small perturbations in input data produce large swings in calibrated parameters — a failure mode PMs care about.

INSIGHT

Why this matters on a rates desk. In PCA-based yield curve risk, each bucket DV01 is projected onto the principal component basis. The level component ( $q_1$ , approximately the flat vector) captures parallel shifts; the slope component ( $q_2$ ) captures steepening/flattening. A trader who hedges only the level exposure while ignoring slope is exposed to a curve twist — which the spectral decomposition makes mathematically explicit.

Assumptions and conventions

Matrices are real and square: $A \in \mathbb{R}^{n \times n}$ .
For the Spectral Theorem, $A$ is symmetric: $A^\top = A$ .
Eigenvectors are normalised to unit length: $\|q\|_2 = 1$ .
Eigenvalues of a real symmetric matrix are real (proved below) and eigenvectors can be chosen orthonormal.
The eigendecomposition $A = Q\Lambda Q^\top$ uses $Q$ orthogonal ( $Q^\top Q = I$ ) and $\Lambda$ diagonal.
numpy.linalg.eigh is used for symmetric matrices (faster, guaranteed real eigenvalues); numpy.linalg.eig is the general (possibly complex) version.

Theory

1. Eigenvalues and Eigenvectors

DEFINITION

Definition 1.1 (Eigenvalue/eigenvector). A scalar $\lambda \in \mathbb{R}$ and non-zero vector $q \in \mathbb{R}^n$ satisfying $Aq = \lambda q$ are called an eigenvalue and its associated eigenvector of $A \in \mathbb{R}^{n \times n}$ .

The equation $Aq = \lambda q$ rewrites as $(A - \lambda I)q = 0$ , which has a non-zero solution iff $A - \lambda I$ is singular, i.e., iff $\det(A - \lambda I) = 0$ .

DEFINITION

Definition 1.2 (Characteristic polynomial). The characteristic polynomial of $A$ is $p_A(\lambda) = \det(A - \lambda I).$ This is a degree- $n$ polynomial in $\lambda$ . Its roots (counting multiplicity in $\mathbb{C}$ ) are the eigenvalues of $A$ .

REMARK

Algebraic vs. geometric multiplicity. The algebraic multiplicity of an eigenvalue $\lambda_0$ is its multiplicity as a root of $p_A$ . The geometric multiplicity is $\dim\ker(A - \lambda_0 I)$ — the dimension of the eigenspace. Always: geometric $\leq$ algebraic. Equality for every eigenvalue is the condition for diagonalisability.

EXAMPLE

Example 1.3 (2×2 covariance matrix). Let $\Sigma = \begin{pmatrix} 4 & 2 \\ 2 & 3 \end{pmatrix}$ .

Characteristic polynomial: $p(\lambda) = (4-\lambda)(3-\lambda) - 4 = \lambda^2 - 7\lambda + 8$ .

Eigenvalues: $\lambda = \frac{7 \pm \sqrt{49 - 32}}{2} = \frac{7 \pm \sqrt{17}}{2}$ , so $\lambda_1 \approx 5.56$ , $\lambda_2 \approx 1.44$ .

Both positive — confirming $\Sigma \succ 0$ (positive eigenvalues $\Leftrightarrow$ positive definite for symmetric matrices, proved via the spectral theorem below).

2. Diagonalisation

DEFINITION

Definition 2.1 (Diagonalisation). A matrix $A \in \mathbb{R}^{n \times n}$ is diagonalisable if there exists an invertible $P \in \mathbb{R}^{n \times n}$ and diagonal $\Lambda$ such that $A = P\Lambda P^{-1}, \qquad \Lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_n).$ The columns of $P$ are the eigenvectors of $A$ .

THEOREM

Theorem 2.2 (Sufficient condition for diagonalisability). If $A$ has $n$ distinct eigenvalues, then $A$ is diagonalisable. Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Proof sketch. Suppose $\sum_{i=1}^k \alpha_i q_i = 0$ with $q_i$ eigenvectors for distinct $\lambda_i$ . Apply $A$ repeatedly and subtract to eliminate terms, eventually showing all $\alpha_i = 0$ . $\square$

REMARK

Distinct eigenvalues are sufficient but not necessary. A matrix with repeated eigenvalues may still be diagonalisable (if geometric multiplicity equals algebraic multiplicity for every eigenvalue) — or it may not (Jordan form is needed in the latter case, but this is rarely relevant in finance practice where matrices are symmetric).

3. The Spectral Theorem

The Spectral Theorem is the cornerstone result for symmetric matrices. It guarantees not just diagonalisability, but orthogonal diagonalisation — the eigenvectors form an orthonormal basis of $\mathbb{R}^n$ .

THEOREM

Theorem 3.1 (Spectral Theorem for real symmetric matrices). If $A \in \mathbb{R}^{n \times n}$ is symmetric ( $A^\top = A$ ), then:

All eigenvalues of $A$ are real.
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
There exists an orthogonal matrix $Q$ (i.e., $Q^\top Q = QQ^\top = I$ ) such that $A = Q\Lambda Q^\top, \qquad \Lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_n), \quad \lambda_i \in \mathbb{R}.$ The columns $q_1, \ldots, q_n$ of $Q$ are orthonormal eigenvectors of $A$ .

Eigendecomposition and the Spectral Theorem

Setup

Where eigendecomposition appears in quant finance

Assumptions and conventions

Theory

1. Eigenvalues and Eigenvectors

2. Diagonalisation

3. The Spectral Theorem

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