Python for Finance, 3rd Edition

Appendix A. Linear Algebra and Optimization Toolkit

This appendix collects linear algebra and optimization recipes that you can reuse across the book. The goal is not to replace a full linear algebra course, but to make the mathematical objects behind portfolio construction, regression, calibration, and risk aggregation explicit and directly implementable in Python.

This appendix focuses on the mathematical tools that turn financial quantities into arrays, matrices, and optimization problems. Vectors and matrices provide the language for returns, weights, exposures, and systems of equations. Inner products, norms, and quadratic forms express portfolio returns, tracking errors, and variances. Decompositions reveal factor structure and numerical stability, optimization routines translate constraints and objectives into implementable portfolio and calibration workflows, and symbolic differentiation with SymPy helps verify gradients before they are coded numerically.

Try This in 60 Seconds

Create two arrays, w = np.array([0.6, 0.4]) and r = np.array([0.03, -0.01]), then evaluate w @ r. The result is the one-period portfolio return, and it should make the meaning of an inner product concrete before the appendix introduces larger matrices.

Vectors, Matrices, and Shapes

A vector is an ordered collection of numbers, and a matrix is a rectangular collection of numbers. In financial applications, vectors often represent asset returns, portfolio weights, factor exposures, or model parameters, ...