Is there existing succinct notation for a matrix $[A]_i$ whose elements are all $0$, except that the $i$th row and $i$th column is given by a particular vector, for example the vector
$$\mathbf{z} = \frac{\mathbf{v} – 2\mathbf{v}_i}{a}$$
I'm trying to avoid having to put in something like
$$
[A]_i
=
\begin{bmatrix}
0 & \ldots & \dfrac{\mathbf{v}_1 – 2\mathbf{v}_i}{a} & 0 & \ldots & 0 \\
\vdots & \ddots & \dfrac{\mathbf{v}_2 – 2\mathbf{v}_i}{a} & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\dfrac{\mathbf{v}_1 – 2\mathbf{v}_i}{a} & \dfrac{\mathbf{v}_2 – 2\mathbf{v}_i}{a} & \ldots & \ldots & \ldots & \dfrac{\mathbf{v}_n – 2\mathbf{v}_i}{a} \\
0 & \ldots & \dfrac{\mathbf{v}_{i+1} – 2\mathbf{v}_i}{a} & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\end{bmatrix}
$$
Best Answer
If ${\bf e_i}$ is the column vector with $1$ in position $i$ and $0$ elsewhere, this is ${\bf z} {\bf e_i}^\top + {\bf e_i} {\bf z}^\top - z_i {\bf e_i} {\bf e_i}^\top$.