Tridiagonal Toeplitz matrix

Occupation pattern of a tridiagonal Toeplitz matrix of size 5 × 5

In linear algebra, a tridiagonal Toeplitz matrix is a tridiagonal matrix with constant main diagonal and secondary diagonal elements . Tridiagonal Toeplitz matrices occur quite frequently in numerical mathematics , for example when calculating cubic splines or when discretizing partial differential equations of the second order in a spatial dimension.

definition

A tridiagonal Toeplitz matrix or is a square matrix of shape ${\displaystyle T\in \mathbb {R} ^{n\times n}}$ ${\displaystyle T\in \mathbb {C} ^{n\times n}}$

${\displaystyle T={\begin{pmatrix}a&c&&0\\b&\ddots &\ddots &\\&\ddots &\ddots &c\\0&&b&a\end{pmatrix}}}$,

where , and are real or complex numbers. A tridiagonal Toeplitz matrix is ​​thus both a special tridiagonal matrix in which the main and secondary diagonal elements are constant, and a special Toeplitz matrix in which the entries outside the main and secondary diagonals are equal to zero . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$

properties

Linear systems of equations of the form can be solved efficiently with the help of the Thomas algorithm , a simplified variant of Gaussian elimination , with an effort of the order . ${\displaystyle Tx=d}$ ${\displaystyle O(n)}$

The eigenvalues and eigenvectors of a real tridiagonal Toeplitz matrix can be given explicitly. If the secondary diagonal entries and are not equal to zero, then the eigenvalues ​​of all are different and through ${\displaystyle T}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle T}$

${\displaystyle \lambda _{k}=a+2{\sqrt {bc}}\cos \left({\frac {\pi k}{n+1}}\right)}$

given with . The associated eigenvectors are ${\displaystyle k=1,\ldots ,n}$

${\displaystyle v^{k}=\left(\left({\frac {b}{c}}\right)^{1/2}\sin \left({\frac {1\pi k}{n+1}}\right),\ldots ,\left({\frac {b}{c}}\right)^{n/2}\sin \left({\frac {n\pi k}{n+1}}\right)\right)}$.

Is , then has the only eigenvalue . The associated eigenvectors are then the unit vectors if is , if is, and if are. The eigenvalues ​​of are then real if and only if holds. The eigenvalues ​​of a triadiagonal Toeplitz matrix are required, for example, for the numerical stability analysis of the Crank-Nicolson method . ${\displaystyle bc=0}$ ${\displaystyle T}$ ${\displaystyle a}$ ${\displaystyle e_{1}}$ ${\displaystyle c\neq 0}$ ${\displaystyle e_{n}}$ ${\displaystyle b\neq 0}$ ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle b=c=0}$ ${\displaystyle T}$ ${\displaystyle bc\geq 0}$

See also

• Block tridiagonal matrix
• Block Toeplitz Matrix

literature

• Albrecht Böttcher , Sergei M. Grudsky: Spectral Properties of Banded Toeplitz Matrices . SIAM, Philadelphia PA 2005, ISBN 0-89871-599-7 , chapter 2.2.
• Silvia Noschese, Lionello Pasquini, Lothar Reichel: Tridiagonal Toeplitz matrices. Properties and novel applications . In: Numerical Linear Algebra with Applications . tape 20 , no. 2 : Special Issue: Inverse Problems Dedicated to Biswa Datta , 2013, ISSN  1070-5325 , p. 302–326 , doi : 10.1002 / nla.1811 .