linear algebra, higher linear algebra
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For natural numbers and and a set , an matrix of elements of is a function from the Cartesian product to .
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 to have an “addition” operation; multiplication of matrices requires to also have a “multiplication” operation. Usually is at least a rig and often a ring or a field.
More generally, for arbitrary sets and we can define an -matrix to be a function . If has some kind of “infinitary sums” as well as finite “products”, then we can also multiply matrices of this sort: e.g. if 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 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 into and natural transformations between such functors.
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
Textbook accounts:
Last revised on June 17, 2024 at 13:38:09. See the history of this page for a list of all contributions to it.