nLab matrix

Definition

For natural numbers $n$ and $m$ and a set $X$, an $n\times m$ matrix of elements of $X$ is a function $M:[n]\times[m]\rightarrow X$ from the Cartesian product $[n]\times[m]$ to $X$.

Often one uses the term in a context where one can add and multiply matrices using matrix calculus. Addition of matrices of the same dimension requires $X$ to have an “addition” operation; multiplication of matrices requires $X$ to also have a “multiplication” operation. Usually $X$ is at least a rig and often a ring or a field.

More generally, for arbitrary sets $A$ and $B$ we can define an $A\times B$-matrix to be a function $A\times B\to X$. If $X$ has some kind of “infinitary sums” as well as finite “products”, then we can also multiply matrices of this sort: e.g. if $X$ is the set of objects of a monoidal category with arbitrary coproducts.

Note that if the structure on X is such that matrix multiplication is associative and there exist identity matrices, then matrices over X may be taken as the morphisms of a category $Mat_X$ whose composition is matrix multiplication. This is in particular the case if X is a field, and so many basic theorems of linear algebra may be understood as concerning functors from $Vect_X$ into $Mat_X$ and natural transformations between such functors.

Related concepts

• matrix multiplication, matrix calculus

• matrix decomposition

• square matrix

• diagonal matrix

• elementary matrix

• block matrix (partitioned matrix)

• row echelon matrix

• rank of a matrix

• inverse matrix

• matrix equivalence

• monomial matrix, permutation matrix

• principal submatrix

• matrix group, matrix Lie group

• unitary matrix. unitary group

• triangular matrix

• Smith normal form

• stochastic matrix

• adjacency matrix

• linear algebra, general linear group, special linear group, matrix mechanics, matrix theory, matrix Hopf algebra, matrix Lie algebra, matrix Lie group, classical Lie group, universal localization, tensor calculus, moment of inertia, eigenvalue, characteristic polynomial (Cayley-Hamilton theorem), spectral curve

Special cases: S-matrix, classical r-matrix, density matrix, hermitian matrix, skew-symmetric matrix, quantum Yang-Baxter matrix, random matrix, skew-symmetric matrix

Operations on/with matrices: transpose matrix, adjoint matrix trace, matrix factorization, Gauss decomposition, Gram-Schmidt process

Determinants and determinant like notions, and special cases: quasideterminant, Berezinian,Jacobian, Pfaffian, hafnian, Wronskian, resultant, discriminant

References

Historical origins:

• Arthur Cayley: A Memoir on the Theory of Matrices, Philosophical Transactions of the Royal Society of London, 148 (1858) 17-37 [jstor:108649]

Textbook accounts:

• Igor R. Shafarevich, Alexey O. Remizov: §2 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]

See also:

• Wikipedia: Matrix (mathematics)