Matrix of ones

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

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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