Notation for vectors whose coordinates are all identical

Question

I wonder if there is an accepted notation for writing a vector (4,4,4,4,4,4,4,4,4,4,4) is a short form. How do I write the vector containing $n$ 4s?

Answer 1

Unfortunately, there is not a great definitive answer for this. Outside of creating your own new notation, you can a few options, depending on how you treat vectors.

A unit vector approach is the easiest and clearest. If ${\bf e}_i$ is the unit vector in the $i$ dimension, then the vector in question equals $$\sum_{i=1}^n4{\bf e}_i$$

With ellipses, you could try $$\underbrace{(4,\cdots,4)}_{\text{$n$ dimensions}}$$ and reinforce this by stating it is contained in $\Bbb{R}^n$. You could also use column or row matrices instead of ordered tuples or delimit the components with angular braces.

On that note about ordered tuples, you could maybe get away with something like this: $\{4\}^n$. Usually, however, that denotes a set in the form of a Cartesian product, so you might try $\langle 4\rangle^n$ instead.

By definition, you could say “let $\bf Y$ denote the $n$-dimensional vector having $1$ for each of its components” (i.e., it is the sum of all the single-dimension unit vectors) and thenceforth reference $4\bf Y$.

Unfortunately (as I’ve asked on this site before) there is no such thing as a widely understood “matrix-builder notation” if you know what I mean.