Matrix of ones

In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one.<ref>Template:Citation.</ref> For example:

<math>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</math>

Some sources call the all-ones matrix the unit matrix,<ref>Template:MathWorld</ref> 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 Template:Math matrix of ones J, the following properties hold:

• The trace of J equals n,<ref>Template:Citation.</ref> and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1.
• The characteristic polynomial of J is <math>(x - n)x^{n-1}</math>.
• The minimal polynomial of J is <math>x^2-nx</math>.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity Template:Math.<ref>Template:Harvtxt; Template:Harvtxt, p. 65.</ref>
• <math> J^k = n^{k-1} J</math> for <math>k = 1,2,\ldots .</math><ref name="timm">Template:Citation.</ref>
• J is the neutral element of the Hadamard product.<ref>Template:Citation.</ref>

When J is considered as a matrix over the real numbers, the following additional properties hold:

• J is positive semi-definite matrix.
• The matrix <math>\tfrac1n J</math> is idempotent.<ref name="timm"/>
• The matrix exponential of J is <math>\exp(\mu J)=I+\frac{e^{\mu n}-1}{n}J</math>

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.<ref>Template:Citation.</ref> 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 <math>(a\cdot b)\cdot (b\cdot c)=b</math>. Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.<ref>Template:Citation</ref>

See also

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