2 Random Vectors and Matrices
Section 1 was conducted in subscripts. That will not scale: the designs ahead have several factors, and regression has an arbitrary number of predictors. This section rewrites the same ideas in matrix form, where a model is a single equation \(\underline {Y}=XB+\underline {\varepsilon }\) regardless of how many terms it contains.
Three tools are assembled. The algebra of matrices, ranks and inverses gives the language. The expectation and variance-covariance of a random vector give the first two moments in matrix form. Differentiating linear and quadratic forms gives the calculus needed to minimise a sum of squares in one step rather than one parameter at a time. The section closes with idempotent matrices and the distribution of quadratic forms — the result that every sum of squares in the course depends on.
2.2 Expectation of Random Vectors and Variance-Covariance
2.3 Derivatives of Linear and Quadratic Forms
2.4 Practice Problems
2.5 Distribution of Quadratic Forms
2.6 Practice Problems
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.