---
tag: M6311DWJ1E
kind: concept
name: Matrix summability method
msc: [40C05]
specializes: [8X25Q4XYMN]
sources:
  - url: https://en.wikipedia.org/wiki/Silverman–Toeplitz_theorem
---

## Summary

A matrix summability method uses an infinite matrix $A = (a_{nk})$ to turn a sequence $(s_k)$ into the weighted sums $\sum_k a_{nk} s_k$.
The method then takes the limit of these sums.
Most classical [[summability methods|8X25Q4XYMN]], such as [[Cesàro summation|N8ZGG7EW6J]], Nörlund means and Euler summation, are of this kind.
Their properties can be read off from the entries of the matrix, as in the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]] and the [[Kojima–Schur theorem|JB8GQBVW6Q]].

## Explanations

### L1 human

Each row of the matrix is a recipe for averaging the terms of the sequence with certain weights.
Row $n$ produces one number, and the method looks at what these numbers do as $n$ grows.
The Cesàro matrix has row $n$ equal to $\frac{1}{n+1}, \dots, \frac{1}{n+1}$ followed by zeros, so it takes plain averages of the first $n + 1$ terms.
Applied to $1, 0, 1, 0, \dots$ it gives averages that tend to $\frac12$.