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:

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

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

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

↑ 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.
1 2 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 .

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[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.

[timm-5] 1 2 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 .