Matrix of ones explained

In mathematics, a matrix of ones or all-ones matrix has every entry equal to one.^[1] Examples of standard notation are given below:

J _{2} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}; J _{3} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}; J _{2,5} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}; J _{1,2} = \begin{pmatrix} 1 & 1 \end{pmatrix}.

Some sources call the all-ones matrix the unit matrix, 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 matrix of ones J, the following properties hold:

• The trace of J equals n,^[2] 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 .^[3]

J ^{k} = n ^{k-1} J for k = 1 , 2 ,\ldots . ^[4]

• J is the neutral element of the Hadamard product.^[5]

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.^[4]

• The matrix exponential of J is

\exp(\mu J) = I + (e ^{\mu} -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.^[6] 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.

See also

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

Notes and References

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3. , p. 65.

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6. .