Determinantioonikoefitsienti

Determinantioonikoefitsienti is a term sometimes used in discussions of linear algebra to denote the coefficient of the first-order change in the determinant of a square matrix when the matrix is perturbed in a specified direction. It is not a standard term in most mathematical references, but it can help describe how determinantal values respond to small perturbations.

Formally, for a square matrix A ∈ F n×n and a perturbation direction B ∈ F n×n, the determinantoonikoefitsienti D(A; B) is defined as the derivative of det(A + εB) with respect to ε at ε = 0:

D(A; B) = d/dε det(A + εB) |_{ε=0}.

Two equivalent ways to compute it are provided by the Jacobi formula. If A is invertible, D(A; B) = det(A) tr(A^{-1} B). In general, D(A; B) = tr(adj(A) B), where adj(A) is the adjugate of A.

Interpretation and properties can be understood as follows: D(A; B) is linear in B and represents how sensitive the determinant is to perturbations in the direction B. When A is invertible, its value scales with det(A) and the trace of A^{-1}B; when A may be singular, the adjugate formula still yields the correct directional derivative.

Applications are found in matrix perturbation theory and numerical linear algebra, where one studies how small changes to a matrix affect its determinant and related invariants. See also: determinant, adjugate, matrix perturbation theory.