# History of circulant matrices for convolution

Discrete linear convolution $$y[k]=h[n] * x[n]=\sum_{i=-\infty}^{\infty} x[i] h[k-i]$$

can be done with circulant matrices with appropriate zero padding. Is anyone aware of the name of the mathematician who proposed convolution by circulant matrices for the first time?

I checked the first use of circulants in the Oxford English Dictionary, but this is associated with determinants (1881). Jeff's famous Earliest Known Words of Mathematics webpage does not list circulants or circulant matrix.

Circulant: Mathematics.

A species of determinant.

1881 W. S. Burnside & A. W. Panton Theory of Equations xi. §129
Here in all the rows the constituents are the same five quantities taken in circular order, a different one standing first in each row. A determinant of this kind is called a circulant.

• The place to look is Thomas Muir's encyclopedic history of determinants, which used to be at a web page I cited here, but no longer exists, at least at its former URL. However, Muir's many volumes are surely available at google-books, archive.org, and possibly other places. (I don't have time to look now.) Nov 20, 2021 at 22:45
• 1) As you have observed, the theory and use of determinants is known to predate by a century and a half the theory of matrices. Certain "phenomenas" have been observed first on determinants, and only later have been re-considered as more "natural" when expressed in the framework of matrices, the previous determinantal context becoming a consequence of the new matricial context. 2) The concept of continuous convolution has emerged very late, in the 1930s... not to speak of its discrete equivalent that I would trace to the 1960s with the birth of Signal Processing. Nov 20, 2021 at 22:46
• @JeanMarieBecker, Do you think Toeplitz could be behind it? What is main use of a circulant matrix besides convolution? Nov 21, 2021 at 5:06
• It's more a theoretical interest than a practical use. For example, circulant matrix $C=(a_0,a_1,... a_{n-1})$ can be expressed as $C=\sum_k a_kN^k$ where $N$ is the nilpotent matrix with ones just above the diagonal, In this way we can express in a matricial ways facts about polynomials modulo n. Nov 21, 2021 at 5:13