m×nMatrix

An m×n matrix, or m by n matrix, is a rectangular array consisting of m rows and n columns of elements drawn from a field, typically the real numbers or complex numbers. It is commonly denoted A or M, with entries a_ij referring to the element in row i and column j (i = 1,…,m; j = 1,…,n). The set of all m×n matrices over a field F is written F^{m×n}. The integers m and n are nonnegative; if either equals zero, the matrix has no elements (an empty matrix).

Elements are indexed by their row and column position, and matrices are often used to represent linear transformations, systems of equations, or data structures. A common shorthand is to denote A ∈ F^{m×n}.

Operations and properties: Matrix addition and subtraction are defined for matrices of the same size, performed elementwise. Scalar multiplication multiplies every entry by a scalar. Matrix multiplication is defined when the number of columns of the left matrix matches the number of rows of the right: if A is m×p and B is p×n, then AB is m×n, with (AB)_ij = sum_{k=1}^p a_ik b_kj. The transpose A^T is the n×m matrix obtained by exchanging rows and columns. The determinant is defined only for square matrices (m = n), and it provides information about invertibility and volume changes under the corresponding linear transformation. The rank of a matrix is the maximum number of linearly independent rows (or columns).

Special cases: The zero matrix has all entries zero and size m×n. The identity matrix I_n is n×n with ones on the diagonal and zeros elsewhere; it acts as the multiplicative identity for square matrices of size n×n. Invertibility requires m = n and det(A) ≠ 0.

Applications: m×n matrices represent linear maps from F^n to F^m, appear in solving linear systems, and model data in science and engineering, including graphics, statistics, and machine learning.