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I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem).

Problem: For equation (2), I think it should be $\textbf{h}^{T}$ instead of $\textbf{h}$ for matrix multiplication to make sense:

enter image description here

Why I think it should be $\textbf{h}^{T}$

$\textbf{x}$ is a Vx1 dimensional vector. $\textbf{W}$ is a VxN dimensional Matrix. Below I re-write equation (1) with dimensions for each vector and matrix:

$$ \textbf{h}_{1xN} = \textbf{x}^{T}_{1xV}\textbf{W}_{VxN} = {\textbf{v}_{w_{I}}}_{1xN}$$

$\textbf{v}^{'}_{w_{j}}$ is a Nx1 dimensional j-th column of matrix $\textbf{W}^{'}$ and $u_{j}$ is a scalar. Therefore rewriting equation (2) with dimensions (you can only multiple when you use $\textbf{h}^{T}$ and not $\textbf{h}$ as shown in figure):

$${u_{j}}_{1x1} = {\textbf{v}^{'}_{w_{j}}}^{T}_{1xN}.\textbf{h}^{T}_{Nx1}$$

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1 Answer 1

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${v'_{w_j}}^T$ and $h$ are both row vectors, as you mention. I think they're using the center dot between them to denote the dot product, which seems ok. If it were matrix multiplication notation, then I think you're right that it would be ${v'_{w_j}}^T h^T$.

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  • $\begingroup$ Wow !! yes. You are correct. I never realized that "dot product" is different than matrix multiplication. If you are using "dot product" you can say $\textbf{x}\cdot\textbf{y}$ where both $\textbf{x}$ and $\textbf{y}$ are 1xN dimensional vectors: math.odu.edu/~bogacki/math316/transp/1_3 $\endgroup$ May 31, 2016 at 3:42

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