Matrix of ones

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

:<math>J_2 = \begin{bmatrix}

1 & 1 \\

1 & 1

\end{bmatrix},\quad

J_3 = \begin{bmatrix}

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, 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, 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 .
<math> J^k = n^{k-1} J</math> for <math>k = 1,2,\ldots .</math>
J is the neutral element of the Hadamard product.

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. 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.

Matrix of ones

Properties

See also

References