Tridiagonal matrix

In linear algebra , a tridiagonal matrix (also known as a three- band matrix ) is a square matrix that only contains entries other than zero in the main diagonal and in the first two secondary diagonals . Tridiagonal matrices occur quite frequently in numerics , for example when calculating cubic splines , when discretizing the second derivative on one-dimensional areas (especially with Sturm-Liouville problems ), when calculating orthogonal polynomials and systems of functions (e.g. when calculating von Bessel functions) and in Krylow's subspace method based on three-part recursions .

definition

A matrix is called tridiagonal if it has the following form: ${\displaystyle T\in \mathbb {C} ^{n\times n}}$

${\displaystyle T={\begin{pmatrix}t_{1,1}&t_{1,2}&0&\dots &0\\t_{2,1}&t_{2,2}&t_{2,3}&\ddots &\vdots \\0&t_{3,2}&\ddots &\ddots &0\\\vdots &\ddots &\ddots &\ddots &t_{n-1,n}\\0&\dots &0&t_{n,n-1}&t_{n,n}\end{pmatrix}}}$

So it applies to everyone . A tridiagonal matrix is ​​called unreduced or irreducible if the elements in the secondary diagonals are all not equal to zero, that is, applies to all . If the main and secondary diagonal entries are constant, then , and , one speaks of a tridiagonal Toeplitz matrix . ${\displaystyle t_{ij}=0}$ ${\displaystyle |i-j|>1}$ ${\displaystyle t_{ij}\not =0}$ ${\displaystyle |i-j|=1}$ ${\displaystyle t_{1,1}=\ldots =t_{n,n}}$ ${\displaystyle t_{1,2}=\ldots =t_{n-1,n}}$ ${\displaystyle t_{2,1}=\ldots =t_{n,n-1}}$

properties

A tridiagonal matrix is ​​both a special case of a band matrix and a Hessenberg matrix . A diagonally dominant tridiagonal matrix is ​​always regular .

Linear systems of equations with a tridiagonal matrix can be efficiently solved with an effort of O (n) . Either with the very fast Thomas algorithm or, in the case of stability problems, with the help of the Gauss method with pivoting. Systems of equations with tridiagonal matrices can therefore be calculated using a direct solver even with comparatively large dimensions.

literature

• Gerhard Sacrifice: Numerical Mathematics for Beginners. An introduction for mathematicians, engineers and computer scientists. 4th revised edition. Vieweg, Braunschweig et al. 2002, ISBN 3-528-37265-6 .

See also

• Triangular matrix
• Bidiagonal matrix
• Block tridiagonal matrix