Mním

Mn×m denotes the set of all n-by-m matrices over a field F. It is commonly written M_{n×m}(F) or M_{n,m}(F). An element A in Mn×m has entries a_{ij} with i = 1,...,n and j = 1,...,m, and the matrix has n rows and m columns.

Mn×m is a vector space over F under standard addition and scalar multiplication, with dimension nm. A canonical basis is the set of matrices E_{ij} that have a 1 in position (i,j) and 0 elsewhere, for i = 1,...,n and j = 1,...,m.

Operations include matrix addition and scalar multiplication performed entrywise. If A ∈ M_{n×m} and B ∈ M_{m×p}, their product AB is defined and lies in M_{n×p}. In particular, when n = m, M_{n×n}(F) forms a unital associative algebra under matrix addition and multiplication.

Key properties include the rank, which is the dimension of the row or column space and satisfies 0 ≤ rank(A) ≤ min(n,m). The transpose maps Mn×m to M_{m×n}. An inner product can be defined, for example ⟨A,B⟩ = trace(A^T B) for real matrices or ⟨A,B⟩ = trace(A^* B) for complex matrices.

Relation to linear maps: every A ∈ Mn×m represents a unique linear transformation T_A: F^m → F^n by T_A(x) = Ax. Conversely, any linear map from F^m to F^n is represented by a matrix in Mn×m with respect to the standard bases.

Size considerations: over a finite field with q elements, Mn×m(F) has q^{nm} elements; over an infinite field, the set is infinite. Notation varies; F^{n×m} is also used when F is understood.