MatrixMN

MatrixMN is a mathematical object representing a rectangular array with M rows and N columns. The entries are elements of a field F, commonly the real numbers R or complex numbers C. The set of all M-by-N matrices over F is denoted F^{M×N}. MatrixMN can represent linear transformations from F^N to F^M, with standard matrix multiplication corresponding to the composition of transformations.

Key operations on MatrixMN include addition and scalar multiplication, defined componentwise. Matrix multiplication is defined when the inner dimensions match: if A ∈ F^{M×N} and B ∈ F^{N×P}, then the product AB ∈ F^{M×P}. The transpose of A, denoted A^T, is in F^{N×M}. The rank of A, rank(A), is the dimension of its image and satisfies 0 ≤ rank(A) ≤ min(M, N). Determinants are defined only for square matrices (M = N), and a matrix is invertible if det(A) ≠ 0, with inverse A^{-1} ∈ F^{N×N}.

Solving linear systems involves MatrixMN through equations of the form Ax = b, with solution methods including Gaussian elimination and LU decomposition. Existence and uniqueness of solutions relate to the ranks of A and augmented matrices.

Additional properties include the existence of left and right inverses in special cases and the concept of pseudoinverses for non-square or singular matrices. MatrixMN serves as a standard representation in many domains, including linear regression, computer graphics, data analysis, and engineering, where rows often correspond to samples and columns to features.

Historically, matrices emerged in 19th-century linear algebra, with the term “matrix” popularized by Cayley. The notation MatrixMN emphasizes the dimensions M and N of the matrix in question.