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:

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

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.

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.
• The characteristic polynomial of J is ${\displaystyle (x-n)x^{n-1}}$.
• The minimal polynomial of J is ${\displaystyle x^{2}-nx}$.
• 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 ${\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J}$

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

• Zero matrix, a matrix where all entries are zero
• Single-entry matrix

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