Tridiagonal Matrix

A square matrix with nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal (i.e., along the subdiagonal and superdiagonal),

 [a_(11) a_(12) 0 0 ... 0 0; a_(21) a_(22) a_(23) ... ... 0 0; 0 a_(32) a_(33) ... ... a_(n-2,n-1) 0; | ... ... ... ... a_(n-1,n-1) a_(n-1,n); 0 0 ... ... ... a_(n,n-1) a_(nn)].

Computing the determinant of such a matrix requires only O(7n) (as opposed to O(n^3/3)) arithmetic operations (Acton 1990, p. 332). Efficient solution of the matrix equation Ax=y for x, where A is a tridiagonal matrix, can be performed in the Wolfram Language using LinearSolve on A, represented as a SparseArray.

See also

Diagonal Matrix, Jacobi Method, Subdiagonal, Superdiagonal

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Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 331-334, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Tridiagonal and Band Diagonal Systems of Equations." §2.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 42-47, 1992.

Referenced on Wolfram|Alpha

Tridiagonal Matrix

Cite this as:

Weisstein, Eric W. "Tridiagonal Matrix." From MathWorld--A Wolfram Web Resource.

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