Linear Algebra for Quantitative Finance › Summary

This chapter covered the linear algebra foundations that underpin quantitative finance. Key concepts include:

• Vectors represent portfolios, returns, and factor exposures. For beginning-of-period weights that sum to one, the dot product with same-period simple returns gives the one-period fully invested portfolio simple return when there are no intra-period cash flows or trading costs. Vector norms measure size and appear in regularization constraints.

• Matrices organize returns over time and across assets. The covariance matrix describes asset relationships, and the quadratic form $\mathbf{w}^T \Sigma \mathbf{w}$ computes portfolio variance for weights fixed over the modeled return distribution and a matching covariance matrix. Matrix multiplication transforms between spaces and aggregates calculations efficiently.

• Systems of linear equations solve hedging problems, factor replication, and regression. Hedging finds instrument quantities to neutralize selected current modeled sensitivities, replication matches target exposures, and regression estimates factor loadings. Least squares handles overdetermined systems by minimizing squared residuals.

• Matrix decompositions expose mathematical patterns in raw data. Eigendecomposition of covariance matrices identifies principal variance directions. PCA reduces dimensionality when only the leading components are retained. SVD provides another route to dominant patterns and pseudoinverses, while Cholesky decomposition enables efficient simulation of correlated variables.

These tools form the computational backbone for the portfolio optimization, factor models, and risk management techniques developed in subsequent chapters. The goal of the derivations and examples is to make later uses of standard library routines easier to inspect and reason about.