# Matrix rows notation

I'm working with a set of $M$ vectors $\{\mathbf{w}_i \in \mathbb{R}^N, \, i = 1, \ldots, M \}$. Since single vectors are usually considered as column vectors, I'm defining a matrix

$$\mathbf{W} = [\mathbf{w}_1, \ldots, \mathbf{w}_M] \in \mathbb{R}^{N \times M}$$

by placing the vectors as matrix columns.

However, for some descriptions, I need to refer to the matrix rows.

Is there an elegant notation to refer to this matrix rows (preferably with less notation overhead)?

• What about $\mathbf{W}_{i\bullet}$ for the $i$th row? – Raskolnikov May 7 '13 at 6:23
• Since I'm writing vectors in lower case I would write matrix rows as $\mathbf{w}_{j*}$ for $j = 1, \ldots, N$. However, I'm not sure if this dot or star symbol is a valid math notation. – Robinaut May 7 '13 at 7:06
• It's very common. But even if it wasn't, you can always invent your own notation, as long as you define it so that readers know. – Raskolnikov May 7 '13 at 7:15

## 3 Answers

You can represent the rows of $\mathbf W$ by the $M$-(column-)vectors $\mathbf w'_i :=\mathrm{col}_i\mathbf W^\mathsf T\;(i=1,\dots,N).$ There is no standard notation for this; I chose the prime notation for convenience. You would also need to state your chosen notation explicitly.

Since $W_{nm}$ is unambiguous for the $n$th row, $m$th col entry of $W$, I quite like the following notation:

\begin{align} \text{For rows:}& \quad W_{n*} = W_{n,*} \\ \text{For cols:}& \quad W_{*m} = W_{*,m} \\ \end{align}

From a programming perspective, this is similar to how in R we use W[n,] and W[,m] and in Numpy we use W[n,:] and W[m,:].

If you write $$\mathbf{W} = [\mathbf{w}_1, \ldots, \mathbf{w}_M]^t \in \mathbb{R}^{M \times N}$$ then the rows of $\mathbf W$ are $\mathbf{w}_1^t, \ldots, \mathbf{w}_M^t$.

• Well, yes, but that's not how $\mathbf{W}$ was written (nor is it even necessarily of the correct dimension), so how does this help? – Cameron Buie May 7 '13 at 6:41
• It's additional notation. Just use a different letter and set it equal to the original. Not that hard. – Jim May 7 '13 at 15:49