identiteedimaatriks

Identiteedimaatriks, often called the identity matrix, is the square matrix I_n with ones on the main diagonal and zeros elsewhere. It is denoted by I_n or simply I and serves as the neutral element for matrix multiplication.

It serves as the multiplicative identity in the ring of n-by-n matrices: for any n-by-n matrix A, AI = IA = A. This makes I_n the identity operator on an n-dimensional vector space, preserving any vector under multiplication: I_n x = x for all vectors x.

Key properties of the identity matrix include det(I_n) = 1, tr(I_n) = n, and rank(I_n) = n. All eigenvalues of I_n are equal to 1, and its minimal polynomial is x − 1. The inverse of the identity matrix is itself: (I_n)^{-1} = I_n. As a representation, I_n corresponds to the identity linear transformation in the standard basis.

In practice, the identity matrix appears whenever an identity operation is needed in linear algebra, such as in solving systems, in expressions involving perturbations, or in the study of eigenvalues and matrix equations. It is the canonical example of a matrix that leaves the structure of a vector space unchanged under linear transformation.