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

↑ Kaare Brandt Petersen & Michael Syskind Pedersen (2008-02-16). "The Matrix Cookbook" (PDF).
↑ 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.
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Examples:
- "Distributed Gain Matrix Optimization in Non-Regenerative MIMO Relay Networks" (PDF).
- Marcel Blattner. "B-Rank: A top N Recommendation Algorithm" (PDF).

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