kertalukuja

Kertalukuja is the plural form of kertaluku, a Finnish mathematical term that denotes multiplicity—the number of times a given value occurs in a mathematical setting. The concept is used to describe repeated roots, repeated eigenvalues, or repeated elements within a multiset or sequence.

In polynomials, the multiplicity of a root α is the largest integer k such that (x − α)^k divides the polynomial. For example, in the polynomial p(x) = (x − 3)^4(x − 1)^2, the root 3 has multiplicity 4 and the root 1 has multiplicity 2. In linear algebra, an eigenvalue λ of a matrix A can have algebraic multiplicity equal to its multiplicity as a root of the characteristic polynomial det(A − λI). The geometric multiplicity of λ, defined as the dimension of the eigenspace corresponding to λ, is always less than or equal to the algebraic multiplicity, and equality holds in some special cases.

Kertalukuja also appears in multiset theory and combinatorics, where it describes how many times an element occurs within a multiset. In numerical contexts, multiplicities can be sensitive to perturbations, so they may change under small changes to the object (for example, eigenvalues can split under perturbation).

See also: algebraic multiplicity, geometric multiplicity, multiplicity (multiset).