# Tridiagonal matrix

2010 Mathematics Subject Classification: *Primary:* 15B05 [MSN][ZBL]

A matrix with non-zero entries only on the main diagonal and the diagonals immediately above and below, for example $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & 0 & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & b_3 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & c_{n-1} & a_n \end{pmatrix} \ . $$

A system of linear equations of the form $Tx = b$, with $T$ tridiagonal, can be solved by a modified form of Gaussian elimination: such systems appear, for example, in the Crank-Nicolson method for numerical solution of non-linear differential equations.

The determinant of a tridiagonal matrix may be computed as a continuant.

See also: Jacobi matrix.

## References

- Thomas Muir.
*A treatise on the theory of determinants*. (Dover Publications, 1960 [1933])

**How to Cite This Entry:**

Tridiagonal matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Tridiagonal_matrix&oldid=43141