Tridiagonal Matrix

A Matrix with Nonzero elements only on the diagonal and slots horizontally or vertically adjacent the diagonal. A general $4\times 4$ tridiagonal Matrix has the form

$$\left[{\matrix{a_{11} & a_{12} & 0 & 0\cr a_{21} & a_{22} & ... ...{32} & a_{33} & a_{34}\cr 0 & 0 & a_{43} & a_{44}\cr}}\right].$$

Inversion of such a matrix requires only $n$ (as opposed to $n^3$) arithmetic operations (Acton 1990).

See also Diagonal Matrix, Jacobi Algorithm

References

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 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.