Single-entry matrix

In mathematics a single-entry matrix is a matrix where a single element is one and the rest of the elements are zero,^[1][2] e.g.,

$\mathbf{J}^{23} = \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}\right].$

It is a specific type of a sparse matrix. The single-entry matrix can be regarded a row-selector when it is multiplied on the left side of the matrix, e.g.:

$\mathbf{J}^{23}\mathbf{A} = \left[ \begin{matrix} 0 & 0& 0 \\ a_{31} & a_{32} & a_{33} \\ 0 & 0 & 0 \end{matrix}\right]$

Alternatively, a column-selector when multiplied on the right side:

$\mathbf{A}\mathbf{J}^{23} = \left[ \begin{matrix} 0 & 0 & a_{12} \\ 0 & 0 & a_{22} \\ 0 & 0 & a_{32} \end{matrix}\right]$

The name, single-entry matrix, is not common, but seen in a few works.^[3]

References

1. Kaare Brandt Petersen & Michael Syskind Pedersen (2008-02-16). "The Matrix Cookbook" (PDF).
2. Shohei Shimizu, Patrick O. Hoyer, Aapo Hyvärinen & Antti Kerminen (2006). "A Linear Non-Gaussian Acyclic Model for Causal Discovery" (PDF). Journal of Machine Learning Research 7: 2003–2030.
3. Examples:
   • "Distributed Gain Matrix Optimization in Non-Regenerative MIMO Relay Networks" (PDF). line feed character in |title= at position 40 (help)
   • Marcel Blattner. "B-Rank: A top N Recommendation Algorithm" (PDF).