Mn×mF

Mn×m(F) denotes the set of all n-by-m matrices with entries from a field F. This collection forms a vector space over F under standard matrix addition and scalar multiplication. Its dimension as an F-vector space is nm, and a convenient basis consists of the nm matrices Eij, where Eij has a 1 in the (i, j) position and zeros elsewhere. Every matrix A in Mn×m(F) can be written uniquely as aij Eij, with A = sum over i=1..n and j=1..m of aij Eij.

Mn×m(F) is naturally identified with the space Hom(F^m, F^n) of linear maps from F^m to F^n, by associating a matrix A with the linear map x ↦ Ax. Under this identification, the columns of A correspond to the images of the standard basis vectors of F^m.

Matrix multiplication is not a binary operation on Mn×m(F) in general; it is defined when the inner dimensions match. Specifically, if A ∈ Mn×p(F) and B ∈ p×m(F), then the product AB is in Mn×m(F). Invertibility is possible only when n = m, in which case the invertible elements form the general linear group GL(n,F). For any matrix in Mn×m(F), its rank ranges from 0 to min(n, m).

If F is the real or complex field, Mn×m(F) carries additional structures, such as the Frobenius inner product ⟨A, B⟩ = trace(AᵀB) (or using the conjugate transpose in C) and the associated Frobenius norm ||A||F = sqrt(trace(AᵀA)). These structures yield standard tools for measuring similarity, closeness, and orthogonality among matrices.

Example: for n = 2 and m = 3, a matrix A = [ [a11, a12, a13], [a21, a22, a23] ] is an element of M2×3(F).