Linear Algebra
6 articles ~23 min
- Covariance and Correlation Matrices Covariance and correlation matrices summarize how asset returns move together. The covariance matrix is the foundation of modern portfolio theory and risk management.
- Eigenvalues and Eigenvectors An eigenvector of a matrix is a direction that doesn't change when the matrix transforms it; only scales. Eigenvalues tell you by how much. They are fundamental to PCA, factor models, and Markov chains.
- Matrix Operations Matrix multiplication is not element-wise; it combines transformations. Understanding how to multiply matrices, invert them, and interpret the results is core to regression and risk models.
- Principal Component Analysis PCA finds the directions of maximum variance in high-dimensional data. It is the workhorse of factor extraction, risk decomposition, and data visualization in quantitative finance.
- Solving Linear Systems and the Matrix Inverse Most problems in quantitative finance, from pricing options to optimizing portfolios, boil down to solving linear systems of equations. Understanding when you can solve Ax = b, and how, is fundamental.
- Vectors and Matrices A vector is a list of numbers; a matrix is a table of numbers. Linear algebra is the language of multi-factor models, portfolio optimization, and machine learning.