Matrisen

Matrisen is the mathematical construct known as a matrix: a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Typically denoted by uppercase letters such as A or M, its entries aij belong to a field, most commonly the real or complex numbers. A matrix is characterized by its dimensions m × n, with m rows and n columns. The transpose flips rows and columns, yielding A^T.

Operations on matrisen include addition and scalar multiplication, performed elementwise. Matrix multiplication is defined when the inner dimensions match: if A is m×n and B is n×p, their product C = AB is m×p with cij = sum over k of aik bkj. The identity matrix In acts as a multiplicative identity. A matrix is invertible if det(A) ≠ 0, with A^-1 being its inverse; the determinant measures a linear transformation’s scale and is defined only for square matrices. The rank is the maximum number of linearly independent rows or columns; row reduction yields row echelon form and can determine the solvability of linear systems.

Eigenvalues and eigenvectors are central concepts: Av = λv for nonzero v, leading to eigen decomposition and, for suitable matrices, diagonalization. The spectral theorem guarantees a real eigenbasis for symmetric matrices. Special matrix classes include diagonal, triangular, symmetric, sparse, and stochastic matrices, each with distinct properties and applications.

Applications of matrisen span solving systems of linear equations, computer graphics and geometric transformations, data analysis and machine learning, Markov chains, and various areas of physics and engineering. History notes that the term matrix was introduced in the mid-19th century by James Sylvester and further developed by Arthur Cayley, leading to its central role in modern mathematics. In Swedish-language texts, matrisen refers to the matrix in general.