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:

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

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 ${\displaystyle (x-n)x^{n-1}}$.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1.[4]
• ${\displaystyle J^{k}=n^{k-1}J}$ for ${\displaystyle k=1,2,\ldots .}$[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 ${\displaystyle {\tfrac {1}{n}}J}$ is idempotent.[5]
• The matrix exponential of J is ${\displaystyle \exp(J)=I+{\frac {e^{n}-1}{n}}J.}$

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 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

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.