Linear Algebra for Quantitative Finance › Linear Algebra for Quantitative Finance

A portfolio return can be a dot product, its variance a quadratic form, and a multi-factor hedge a system of equations. The same small set of linear-algebra operations turns hundreds of asset-level quantities into calculations you can inspect and implement. Eigenvectors of a covariance matrix then provide directions of variation that can be used to study common movements in returns.

This chapter builds your fluency in linear algebra with a constant eye toward financial applications. We start with vectors and matrices, showing how they naturally represent portfolios and return data. We then tackle systems of linear equations through factor replication, hedging, and regression examples. Finally, we explore matrix decompositions, the techniques that enable principal component analysis and expose patterns in financial data.