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Matrix

A rectangular array of numbers arranged in rows and columns -- the central algebraic object used throughout statistics to represent data tables, linear transformations, and systems of equations compactly.

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In Plain English

A matrix is a grid of numbers, and in statistics that grid is almost always your data: rows for observations, columns for variables. But a matrix is more than a storage format, it's an algebraic object you can multiply, invert, and decompose, which is exactly what lets a formula like ordinary least squares be written and computed as one compact matrix expression instead of a page of separate equations, one per variable.

Definition

A matrix $A$ is a rectangular array of numbers arranged into $m$ rows and $n$ columns, written $A \in R^{m \times n}$ , with entry $a_{ij}$ denoting the number in row $i$ , column $j$ . In statistics a matrix most commonly stores a full dataset (rows as observations, columns as variables, the design matrix of a regression), a set of pairwise relationships among variables (a covariance or correlation matrix), or a linear transformation (multiplying a vector by a matrix maps it to a new vector), and virtually every multivariate statistical method -- regression, PCA, MANOVA, mixed models -- is expressed and computed as an operation on one or more matrices.

Formula

A = a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 n} ⋮ a_{mn}

Notation

a_{ij}

—entry in row i, column j

m \times n

—the matrix's dimensions (m rows, n columns)

Properties

A matrix is square only when its number of rows equals its number of columns -- squareness is a prerequisite for several other concepts in this category (determinant, inverse, eigenvalues, symmetric/orthogonal/positive-definite matrices), which is why so much of matrix-based statistical theory revolves around square matrices like $X^{'} X$ built from a rectangular design matrix $X$ .
Matrix addition and scalar multiplication work entrywise and require matching dimensions, but matrix multiplication (see Matrix Multiplication) is a fundamentally different, non-commutative operation that requires the first matrix's column count to match the second matrix's row count -- a dimension-compatibility rule that governs every valid matrix expression in statistical formulas.
In applied statistics, the design matrix $X$ (rows = observations, columns = predictors) is the single most common matrix object, since ordinary least squares, ANOVA, and virtually every GLM are all fit by solving matrix equations built from $X$ and the response vector $y$ .

At a Glance

Written asan m x n rectangular grid of numbers

Common statistical rolesdata table, design matrix, covariance/correlation matrix, linear map

Square matrixspecial case where m = n, required for a determinant, inverse, or eigenvalues

Core operationsaddition, multiplication, transpose, inverse (if square and invertible)

Last updated September 8, 2026← Back to the Dictionary