A matrix is a rectangular grid of numbers, arranged in rows and columns, that represents a linear transformation between vector spaces. Matrices let you compactly represent and efficiently compute with systems of linear equations, geometric transformations, and much more — making them one of the most practically important tools in all of applied mathematics.
A matrix is simply a rectangular table of numbers, like a small spreadsheet:
But a matrix isn't just a static table — it represents an action: a way of transforming points in space. The specific matrix above, for example, stretches every point twice as far horizontally and three times as far vertically.
Matrices are the computational heart of linear algebra — arguably the single most widely applied branch of mathematics across science, engineering, economics, and computing, precisely because so many real-world relationships can be usefully approximated or exactly modelled as linear.
Multiplying a matrix by a vector transforms that vector. Multiplying two matrices together corresponds to composing their two transformations — doing one transformation, then the other, in sequence. Matrix multiplication is associative, (AB)C = A(BC), but generally not commutative: AB ≠ BA in general, reflecting that the order in which you apply transformations usually matters.
The identity matrix I (with 1s down the diagonal, 0s elsewhere) represents "do nothing" — leaving every vector unchanged. A matrix A is invertible if there exists A⁻¹ such that AA⁻¹ = A⁻¹A = I — meaning the transformation A can be perfectly undone. Not every matrix has an inverse; a matrix fails to be invertible exactly when its determinant is zero.
Every linear transformation T : ℝⁿ → ℝᵐ can be represented by an m×n matrix, once you fix a choice of basis for each space (see the Vector Space entry). This is why matrices are so ubiquitous: they are the concrete, computable representation of the abstract notion of a linear map.
| Type | Property | Significance |
|---|---|---|
| Symmetric | A = Aᵀ (equals its transpose) | Always has real eigenvalues |
| Orthogonal | AᵀA = I | Preserves lengths and angles (rotations, reflections) |
| Diagonal | Nonzero only on the diagonal | Represents independent scaling along each axis |
| Hermitian | Complex analogue of symmetric | Represents observables in quantum mechanics |
A system of linear equations can be written compactly as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of results. If A is invertible, the unique solution is x = A⁻¹b. Gaussian elimination — systematically simplifying the matrix using row operations — is the standard practical method for solving such systems, whether or not A is invertible.
Systems of linear equations and array-like arrangements of coefficients were studied long before the modern matrix concept — Chinese mathematicians used array-based methods equivalent to Gaussian elimination in The Nine Chapters on the Mathematical Art (c. 200 BCE–200 CE) to solve systems of linear equations. The term "matrix" itself was coined by James Joseph Sylvester in 1850, and the systematic algebraic theory of matrices (multiplication, inversion) was developed by Arthur Cayley in the 1850s–1860s, who is credited with founding matrix theory as a distinct branch of algebra.
For many matrices, there exist special vectors (eigenvectors) that are simply scaled (not rotated or otherwise distorted) by the transformation, by a factor called the corresponding eigenvalue. When a matrix has a full set of linearly independent eigenvectors, it can be "diagonalised" — rewritten in a coordinate system where the transformation acts as simple independent scaling along each axis. This decomposition is fundamental to understanding a matrix's long-term behaviour under repeated application, and underlies techniques from vibration analysis in engineering to Google's original PageRank algorithm (which finds the dominant eigenvector of the web's link matrix).
Every matrix (even non-square, non-invertible ones) admits a Singular Value Decomposition, A = UΣVᵀ, factoring it into a rotation, a scaling, and another rotation. The SVD is one of the most computationally useful and numerically stable matrix decompositions, with applications ranging from data compression and noise reduction to the recommendation algorithms used by streaming services (via a related technique called matrix factorisation) and modern large language model architectures.
The straightforward algorithm for multiplying two n×n matrices takes O(n³) operations. Volker Strassen (1969) discovered an algorithm requiring only about O(n^2.807) operations — a genuinely surprising result showing that the "obvious" method is not actually optimal. Subsequent research (continuing actively into the 2020s, including notable contributions using AI-assisted search methods by DeepMind's AlphaTensor in 2022) has pushed the theoretical exponent even lower, though the currently best-known practical algorithms remain a highly active area of ongoing research, and the true optimal exponent remains an open problem.