Difference between revisions of "Tridiagonal matrix"
(Start article: Tridiagonal matrix) |
(MSC 15B05) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | {{TEX|done}}{{MSC|15B05}} | ||
| + | |||
A matrix with non-zero entries only on the main diagonal and the diagonals immediately above and below, for example | A matrix with non-zero entries only on the main diagonal and the diagonals immediately above and below, for example | ||
$$ | $$ | ||
| Line 13: | Line 15: | ||
The [[determinant]] of a tridiagonal matrix may be computed as a [[continuant]]. | The [[determinant]] of a tridiagonal matrix may be computed as a [[continuant]]. | ||
| + | |||
| + | See also: [[Jacobi matrix]]. | ||
==References== | ==References== | ||
* Thomas Muir. ''A treatise on the theory of determinants''. (Dover Publications, 1960 [1933]) | * Thomas Muir. ''A treatise on the theory of determinants''. (Dover Publications, 1960 [1933]) | ||
| − | |||
| − | |||
Latest revision as of 18:31, 14 April 2018
2020 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])
Tridiagonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tridiagonal_matrix&oldid=37470