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The category of matrices is equivalent to the category of finite-dimensional real vector spaces and linear maps. This is witnessed by the functor mapping the number n {\displaystyle n} to the vector space R n {\displaystyle \mathbb {R} ^{n}} , and an n × m {\displaystyle n\times m} matrix to the corresponding linear map R m → R n ...
A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices.
This category has the following 3 subcategories, out of 3 total. R. Random matrices (20 P) S. Sparse matrices (19 P) Σ. Matrix stubs (59 P) Pages in category "Matrices"
Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring. [8] In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers.
In the language of Category theory, the mixed-product property of the Kronecker product (and more general tensor product) shows that the category Mat F of matrices over a field F, is in fact a monoidal category, with objects natural numbers n, morphisms n → m are n×m matrices with entries in F, composition is given by matrix multiplication ...
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories.
Matrix theory is a branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra , but soon grew to include subjects related to graph theory , algebra , combinatorics and statistics .