Can the dot of the dot product be omitted?

In arithmetic, the symbol for multiplication, "$\cdot$" or "$\times$", is usually omitted for letters, right? (e.g., $y = a + bx$ rather than $y = a + b \cdot x$ or $y = a + b \times x$).

Can we do the same for the dot product of two vectors in a data frame? For example, $y_i = \mathbf{x}_{i}\mathbf{\beta} = \beta_1x_{{1}_{i}} + \beta_2x_{{2}_{i}}...$ rather than $y_i = \mathbf{x}_{i}\cdot\mathbf{\beta}$, where $i$ is the index of rows in the data frame, so $\mathbf{x}_{i}$ is the $i$th row of the data and therefore a vector consisting of $x_{{1}_{i}}, x_{{2}_{i}}...$; $x_{{j}_{i}}$ is the $j$th variable in $\mathbf{x}_{i}$ for the $i$th row of the data; $\mathbf{\beta}$ is a vector of parameters for each variable, $x_{{1}_{i}}, x_{{2}_{i}}...$.

As far as I searched on the Internet, I could not find any explanation to indicate the possibility of omitting the dot in the dot product. I know in matrix notation, there is usually no symbol between multiplied matrices (e.g., $\mathbf{AB}$), but I'm not sure if this applies to vector multiplication.

I would appreciate your insight!

Solution 1:

It is only a matter of convention. Indeed, in different parts of mathematics and science, there are a lot of very specialized notations, often ommitting many symbols for the sake of brevity.

If you write a vector-vector product, there are plenty of different things that could mean

• inner product, denoted as:
  • $v\cdot w$ or $(v,w)$ (more common in math)
  • $\langle v,w\rangle$ or $\langle v | w\rangle$ (more common in physics)
  • $v^T w$ (or $v^t w$ or $v^\dagger w$, if you use complex numbers). This is essentially a matrix-matrix product, where $v^T$ is a $1\times n$ matrix and $v$ is a $n\times 1$ matrix.
• outer product (or "Kronecker product"), usually denoted as
  • $v\otimes w$ (general notation of tensor product)
  • $v \wedge w$ (this is the anti-symmetric version of the tensor product, usually called "exterior prodcut")
  • $vw^T$ (this is again a matrix-matrix product, very explicitly)
• component-wise product (or "Hadamard product")
  • denoted as $v\circ w$ or $v:w$
• cross-product $v\times w$
  • this is actually the least likely to be written as "$vw$".

The point is, depending on the problem you are solving, any of these might be the "obvious" thing you meant. In fact, I personally have seen all of these written simply as $vw$, in widely different fields of mathematics and science.

So: As long as you (and anybody reading your work) is very clear on what you mean, and nobody is going to waste any time and mental work on figuring out what you meant, then feel free to just use "$vw$". In all other cases, (and mostly if you are unsure) please be more explicit.