Rm×k

R^{m×k} denotes the set of all m-by-k matrices with real entries. An element of R^{m×k} can be represented as an array (a_{ij}) with 1 ≤ i ≤ m and 1 ≤ j ≤ k, where each a_{ij} is a real number. The notation is commonly used in linear algebra, numerical analysis, and applied fields that manipulate rectangular arrays of real-valued data.

R^{m×k} is a real vector space of dimension m·k under entrywise addition and scalar multiplication. A standard basis is given by the matrices E_{ij} that have a 1 in the (i,j) position and 0 elsewhere. When m = k the set becomes the algebra of n-by-n square matrices, which supports matrix multiplication and has additional structure such as determinants and traces.

Common operations include matrix addition, scalar multiplication, transpose (which maps R^{m×k} to R^{k×m}), and, where dimensions permit, matrix multiplication. Inner products and norms are often defined via the Frobenius inner product ⟨A,B⟩ = trace(A^T B), inducing the Frobenius norm. Rank, row and column spaces, and singular value decomposition are central structural concepts.

Topologically, R^{m×k} is isomorphic to Euclidean space R^{m·k} and inherits its standard topology and differentiable structure, making it a natural setting for optimization, statistics, machine learning, and the representation of linear maps from R^k to R^m.