nLab matrix




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

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 XX to have an “addition” operation; multiplication of matrices requires XX to also have a “multiplication” operation. Usually XX is at least a rig and often a ring or a field.

More generally, for arbitrary sets AA and BB we can define an A×BA\times B-matrix to be a function A×BXA\times B\to X. If XX has some kind of “infinitary sums” as well as finite “products”, then we can also multiply matrices of this sort: e.g. if XX 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 XMat_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 XVect_X into Mat XMat_X and natural transformations between such functors.

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


Textbook accounts:

Last revised on September 5, 2023 at 14:59:48. See the history of this page for a list of all contributions to it.