# Matrix of ones

In mathematics, a **matrix of ones** or **all-ones matrix** is a matrix where every element is equal to one.[1] Examples of standard notation are given below:

Some sources call the all-ones matrix the **unit matrix**,[2] but that term may also refer to the identity matrix, a different matrix.

## Properties

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

- The trace of
*J*is*n*,[3] and the determinant is 1 if*n*is 1, or 0 otherwise. - The characteristic polynomial of
*J*is . - The rank of
*J*is 1 and the eigenvalues are*n*with multiplicity 1 and 0 with multiplicity*n*− 1.[4] - for [5]
*J*is the neutral element of the Hadamard product.[6]

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

*J*is positive semi-definite matrix.- The matrix is idempotent.[5]
- The matrix exponential of
*J*is

## 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 a *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*.[7] 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.

## References

- 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. - Weisstein, Eric W. "Unit Matrix".
*MathWorld*. - Stanley, Richard P. (2013),
*Algebraic Combinatorics: Walks, Trees, Tableaux, and More*, Springer, Lemma 1.4, p. 4, ISBN 9781461469988. - Stanley (2013); Horn & Johnson (2012), p. 65.
- Timm, Neil H. (2002),
*Applied Multivariate Analysis*, Springer texts in statistics, Springer, p. 30, ISBN 9780387227719. - Smith, Jonathan D. H. (2011),
*Introduction to Abstract Algebra*, CRC Press, p. 77, ISBN 9781420063721. - Godsil, Chris (1993),
*Algebraic Combinatorics*, CRC Press, Lemma 4.1, p. 25, ISBN 9780412041310.