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In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one. For example:

${\displaystyle J_{2}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},\quad J_{3}={\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}},\quad J_{2,5}={\begin{bmatrix}1&1&1&1&1\\1&1&1&1&1\end{bmatrix}},\quad J_{1,2}={\begin{bmatrix}1&1\end{bmatrix}}.\quad }$

Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different type of matrix.

A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with unit vectors.

Properties

For an n × n matrix of ones J, the following properties hold:

• The trace of J equals n, and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1.
• The characteristic polynomial of J is ${\displaystyle (x-n)x^{n-1}}$.
• The minimal polynomial of J is ${\displaystyle x^{2}-nx}$.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1.
• ${\displaystyle J^{k}=n^{k-1}J}$ for ${\displaystyle k=1,2,\ldots .}$
• J is the neutral element of the Hadamard product.

When J is considered as a matrix over the real numbers, the following additional properties hold:

• J is positive semi-definite matrix.
• The matrix ${\displaystyle {\tfrac {1}{n}}J}$ is idempotent.
• The matrix exponential of J is ${\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J}$

Applications

The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA. As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.

The logical square roots of a matrix of ones, logical matrices whose square is a matrix of ones, can be used to characterize the central groupoids. Central groupoids are algebraic structures that obey the identity ${\displaystyle (a\cdot b)\cdot (b\cdot c)=b}$. Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.

See also

• Zero matrix, a matrix where all entries are zero
• Single-entry matrix

References

1. Horn, Roger A.; Johnson, Charles R. (2012), "0.2.8 The all-ones matrix and vector", Matrix Analysis, Cambridge University Press, p. 8, ISBN 9780521839402.
2. Weisstein, Eric W., "Unit Matrix", MathWorld
3. Stanley, Richard P. (2013), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, Lemma 1.4, p. 4, ISBN 9781461469988.
4. Stanley (2013); Horn & Johnson (2012), p. 65.
5. ^ Timm, Neil H. (2002), Applied Multivariate Analysis, Springer texts in statistics, Springer, p. 30, ISBN 9780387227719.
6. Smith, Jonathan D. H. (2011), Introduction to Abstract Algebra, CRC Press, p. 77, ISBN 9781420063721.
7. Godsil, Chris (1993), Algebraic Combinatorics, CRC Press, Lemma 4.1, p. 25, ISBN 9780412041310.
8. Knuth, Donald E. (1970), "Notes on central groupoids", Journal of Combinatorial Theory, 8: 376–390, doi:10.1016/S0021-9800(70)80032-1, MR 0259000

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