Enhetsmatrisen

Enhetsmatrisen, also known as the identity matrix in English, is a square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I_n for an n-by-n matrix. As the multiplicative identity of matrix multiplication, multiplying any compatible matrix A by I_k (where k is the number of columns of A) from the left or right leaves A unchanged: I_k A = A and A I_m = A.

The elements of an identity matrix are defined as δ_{ij}, where δ_{ij}=1 if i=j and 0 otherwise. This Kronecker delta notation underlines the matrix's role in linear algebraic operations, including solving linear systems, computing inverses, and performing change‑of‑basis transformations.

Identity matrices appear in many contexts: they represent the unaltered linear transformation in vector spaces, serve as the neutral element in ring theory, and provide a foundational block in constructing block diagonal matrices. In computational applications, identity matrices are used as initial guesses for iterative solvers and as scaling matrices.

The concept dates back to early 20th‑century linear algebraists who formalized matrix theory. While the Swedish term “enhetsmatrisen” is used mainly in Scandinavian academic literature, the English “identity matrix” and German “Einheitsmatrix” are common in international texts. The notation I is widespread across mathematics, physics, engineering, and computer science, making it one of the most universally recognized symbols.