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The set M(n, R) (also denoted M n (R) [7]) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module R n. [58] If the ring R is commutative , that is, its multiplication is commutative, then the ring M( n , R ) is also an associative algebra over R .
An M-matrix is commonly defined as follows: Definition: Let A be a n × n real Z-matrix.That is, A = (a ij) where a ij ≤ 0 for all i ≠ j, 1 ≤ i,j ≤ n.Then matrix A is also an M-matrix if it can be expressed in the form A = sI − B, where B = (b ij) with b ij ≥ 0, for all 1 ≤ i,j ≤ n, where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an ...
A number of matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a m-by-n matrix A and a n-by-k matrix B is the m-by-k matrix C given by (), = =,,. [2] This matrix product is denoted AB.
From the original definition, the matrix presents n data points in an m-dimensional space. The L 2 , 1 {\displaystyle L_{2,1}} norm [ 6 ] is the sum of the Euclidean norms of the columns of the matrix:
A stochastic matrix is a real matrix of nonnegative entries, such that the sum of each column is one. Stochastic matrices include the identity and are closed under composition, and so they form a subcategory of M a t R {\displaystyle \mathbf {Mat} _{\mathbb {R} }} .
The n × n matrices that have an inverse form a group under matrix multiplication, the subgroups of which are called matrix groups. Many classical groups (including all finite groups ) are isomorphic to matrix groups; this is the starting point of the theory of group representations .
The column space of an m × n matrix with components from is a linear subspace of the m-space. The dimension of the column space is called the rank of the matrix and is at most min(m, n). [1] A definition for matrices over a ring is also possible. The row space is defined similarly.
In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let M(n,m) denote the space of real n×m matrices with n rows and m columns. Such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a column vector, is denoted with a boldface lowercase letter: a, x, y, etc.