Home Statistical Dictionary Block Matrix
Block Matrix
A matrix partitioned into rectangular sub-matrices (blocks), allowing matrix algebra to be carried out on the blocks themselves -- the natural structure for a design matrix with grouped predictors or a covariance matrix with distinct groups of variables.
In Plain English
A block matrix is just an ordinary matrix that's been carved up into smaller rectangular chunks, each chunk (block) treated as if it were a single entry for the purposes of addition and multiplication. This isn't just notational convenience, statistical models with grouped structure, like a design matrix that stacks separate blocks for a fixed-effects part and a random-effects part, are naturally written and manipulated as block matrices, and many mixed-model and multivariate formulas are most cleanly expressed and derived in block form.
Definition
A block matrix (or partitioned matrix) is a matrix divided into rectangular sub-matrices, called blocks, by a grid of horizontal and vertical partition lines, e.g. , where are themselves matrices of compatible dimensions. Block matrix addition and multiplication follow the same rules as ordinary scalar-entry matrix algebra, applied to the blocks (provided the block dimensions are conformable), which lets a large matrix computation be broken into smaller, more manageable sub-computations, and this structure appears throughout statistics in the design matrices of models with grouped or hierarchical predictors, in the joint covariance matrix of two sets of variables (partitioned into within-group and between-group blocks), and in the derivation of formulas such as the block matrix inversion formula used in updating regression estimates as new predictors are added.
Formula
Notation
Properties
- Block matrix multiplication follows exactly the same pattern as ordinary matrix multiplication, with each block treated as if it were a single number, provided the individual blocks' dimensions are conformable for the multiplications and additions the formula requires -- this lets large matrix computations be decomposed into smaller, more tractable sub-computations.
- A joint covariance matrix for two groups of variables is naturally written in 2x2 block form, with diagonal blocks giving each group's own covariance structure and off-diagonal blocks giving the cross-covariances between the two groups -- this partitioned structure underlies formulas for conditional distributions within the multivariate normal family and for canonical correlation analysis.
- Block matrix inversion formulas (expressing the inverse of a partitioned matrix in terms of its individual blocks and their Schur complements) let a regression's coefficient estimates be updated efficiently when a new predictor is added, without refitting the entire model or inverting the full, larger matrix from scratch.
At a Glance
Related Terms