# Toeplitz matrix

$T$- matrix

An infinite matrix $( a _ {nk} )$, $n, k = 1, 2 \dots$ satisfying the conditions:

$$\sum _ {k = 1 } ^ \infty | a _ {nk} | \leq M,\ \ n = 1, 2 \dots$$

where $M$ does not depend on $n$;

$$\lim\limits _ {n \rightarrow \infty } \ a _ {nk} = 0,\ \ k = 1, 2 , . . . ;$$

$$\lim\limits _ {n \rightarrow \infty } \sum _ {k = 1 } ^ \infty a _ {nk} = 1.$$

These conditions are necessary and sufficient for regularity (cf. Regular summation methods) of the matrix summation method defined by sending a sequence $\{ s _ {n} \}$ to a sequence $\{ \sigma _ {n} \}$ via the matrix $( a _ {nk} )$:

$$\sigma _ {n} = \ \sum _ {k = 1 } ^ \infty a _ {nk} s _ {k} .$$

The necessity and sufficiency of these conditions for regularity were proved by O. Toeplitz in the case of triangular matrices.

How to Cite This Entry:
Toeplitz matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_matrix&oldid=49627
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article