tridiagonaalse

Tridiagonaalse is the Dutch term used to describe objects that are tridiagonal. In mathematics, a tridiagonal matrix is a square matrix in which all entries are zero except on the main diagonal and the diagonals immediately above and below it.

If A is an n×n matrix, its main diagonal contains the elements b1, …, bn; the subdiagonal contains the elements a1, …, a_{n-1} in positions (i+1, i); and the superdiagonal contains the elements c1, …, c_{n-1} in positions (i, i+1). All other entries are zero. This compact structure makes tridiagonal matrices a special case of banded matrices with bandwidth 1.

Special cases include a diagonal matrix when all a_i and c_i are zero, a symmetric tridiagonal matrix when a_i = c_i for all i, and a Toeplitz tridiagonal matrix when the nonzero off-diagonal elements are constant and the main diagonal is constant as well. The tridiagonal form often arises from discretizations and structured models.

Computational properties are central to their appeal. Solving Ax = d can be done efficiently with the Thomas algorithm (also known as the tridiagonal matrix algorithm), which operates in linear time O(n). LU decomposition exists for tridiagonal matrices with L and U retaining tridiagonal form, facilitating repeated solves and stability considerations. Eigenvalue computations for tridiagonal matrices are also efficient and widely used in numerical linear algebra.

Applications of tridiagonal matrices appear in numerical methods for differential equations (notably the one-dimensional Poisson equation), finite difference methods, quantum mechanics, vibration analysis, and certain stochastic processes. See also tridiagonal matrix and related numerical algorithms.