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.

A **vector of ones** or **all-ones vector** is matrix of ones having row or column form.

## Properties

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

- The trace of
*J*equals*n*,^{[3]}and the determinant equals 0 for*n*≥ 2, but equals 1 if*n*= 1 (or if*n*= 0 if we want to consider the empty square matrix, which is an all-ones matrix). - 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 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*.^{[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.

## See also

- Zero matrix, a matrix where all elements are zero
- Single-entry matrix

## 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.- ^
^{a}^{b}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.