# Is matrix notation with dots acceptable for papers? What are some alternatives?

I am very used to this kind notation when dealing with matrices with arbitrary dimensions: $$X_P^T\Lambda X_P= \begin{bmatrix} x_1 & x_2, &\dots, &x_n\\ \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & \dots & 0\\ 0 & \lambda_2 & \dots & 0\\ \vdots & \vdots &\ddots \\ 0 & 0 & \dots & \lambda_n\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots\\ x_n\\ \end{bmatrix}$$

I like it because it's nice and directly conveys the matrix representation. However, someone mentioned to me that these representations are good for notes and teaching materials but not for publications. In this specific example the matrix is a diagonal matrix, but I am asking more in general, if you have to convey an arbitrary matrix with any dimensions and any entries, how do you convey the pattern in the matrix without doing dot notation?

• it's perfectly fine for papers Commented May 16, 2020 at 4:52
• I think its perfectly fine. In fact I would even not bother writing out the zeros, but maybe that's just me. If you want to do it without dots, then you can write it as $$\Lambda_{ij} = \delta_{ij} \lambda_i$$ (no Einstein summation) Commented May 16, 2020 at 4:54
• It's fine, if it's the easiest way to explain the form of the matrix. However, if every time you do a matrix calculation you write out the matrix in this form it will take up a lot of page real estate for simple concepts. In this case for example, I would just say "Let $\Lambda$ be a diagonal matrix with entries $\lambda_i$" which is more compact and just as informative. Commented May 16, 2020 at 4:55
• Also, mathematicians often like things to be coordinate-free, so they speak in terms of linear transformations without writing down the matrix if possible. Commented May 16, 2020 at 4:58
• Any reputable journal will have a style guide which tells you how to present the materials for publishing. This varies from journal to journal, just consult them. Commented May 16, 2020 at 5:11

The description on the right is very common and serves to give the reader a schematic description of the matrices.

I would write the equation above as follows, although this is my preference: $$$$x^\top \Lambda x = \begin{bmatrix} x_1 & \cdots & x_n \end{bmatrix} \begin{bmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \end{bmatrix} \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}, \tag{1} \label{myeq}$$$$ where it is understood that the zeros occupy the blank spaces above and below the diagonal.

For $$n \in \mathbb{N}$$, let $$\langle n \rangle := \{ 1, \dots, n\}$$. If $$A:\langle m \rangle \times \langle n \rangle \to \mathbb{F}$$ is a function and $$\mathbb{F}$$ is a set (typically a field), then $$A$$ is called an $$m$$-by-$$n$$ matrix. By convention, the value $$A(i,j)$$ is abbreviated to $$A_{ij}$$ or $$a_{ij}$$.

Given the nature of the function, it is often convenient to represent a matrix via a rectangular array. For instance, the matrix $$A=\{a_{11},a_{12},a_{21},a_{22}\}$$ is written as $$$$A= \begin{bmatrix} a_{11} & a_{12 } \\ a_{21} & a_{22} \end{bmatrix}.$$$$

In general, it is understood that $$A=[a_{ij}] \in \textsf{M}_{m \times n}(\mathbb{F})$$ is the function $$\{a_{ij}\}_{i\in\langle m \rangle, j \in \langle n \rangle}$$.

It is customary in linear algebra/matrix theory journals to define a matrix and give its schematic description as in \eqref{myeq}.

Here is an example from a 2017 paper I co-authored with Charles R. Johnson, who is widely considered to be the best matrix theorist in the world:

I have published and refereed in the top journals in matrix theory and linear algebra and I have never objected to or had a referee object to the practice you describe in your post.

• This answer is perhaps missing some of the info in the comments, like that it;s usually fine for publications but you need to check with the specific journal. As well as suggesting alternative ways to write matrices. Commented May 17, 2020 at 1:16
• @Makogan I simply answered the question you posed. I’ve published and refereed for almost every major journal in matrix theory/linear algebra and what you wrote has never been a problem. Commented May 17, 2020 at 2:22
• Yes I know, I am not saying the answer is bad, I am just suggesting that it could be a way to make the answer more general, to help others that may end up finding this post. :) Commented May 17, 2020 at 2:34