tridiagonale

Tridiagonal, or tridiagonale in Italian, refers to a square matrix in which all nonzero elements lie on the main diagonal and on at most the diagonals immediately above and below it. In other words, every row has nonzero entries only in columns i−1, i, and i+1. An n-by-n tridiagonal matrix thus has at most 3n−2 nonzero elements; a symmetric tridiagonal matrix has at most 2n−1.

The three diagonals are usually described as the subdiagonal (below the main diagonal), the main diagonal, and the superdiagonal (above the main diagonal). Denoting these by a (length n−1), b (length n), and c (length n−1), the entries satisfy A[i,i] = b[i], A[i,i−1] = a[i−1], A[i−1,i] = c[i−1]. In the symmetric case, the sub- and superdiagonals are equal (a = c).

Tridiagonal matrices arise naturally from the discretization of one-dimensional problems, such as the finite-difference approximation of second-order differential operators (for example, the discretized Laplacian). They also appear in chain models and other 1D problems in physics and engineering. A common example is the symmetric Toeplitz tridiagonal matrix with main diagonal 2 and off-diagonals −1, which represents the standard 1D Poisson operator under Dirichlet boundary conditions.

Algorithms exploit the structure of tridiagonal matrices. The Thomas algorithm solves Ax = b in O(n) time for suitable A (e.g., diagonally dominant or positive definite). Eigenvalue problems for tridiagonal matrices benefit from specialized QR and divide-and-conquer methods, yielding efficient computation compared with dense approaches.

Variants include general, symmetric, Toeplitz, and bordered tridiagonal forms, each with particular structural features and applications in numerical linear algebra and scientific computing.