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In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one.^[1] For example:

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,^[2] 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 $n \times n$ 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.
The characteristic polynomial of J is $(x-n)x^{n-1}$ .
The minimal polynomial of J is $x^{2}-nx$ .
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 ${\tfrac {1}{n}}J$ is idempotent.^[5]
The matrix exponential of J is $\exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J$

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.^[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.

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 $(a\cdot b)\cdot (b\cdot c)=b$ . Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.^[8]

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.
^ 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

[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] 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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Properties

Applications

See also

References