 #jsDisabledContent { display:none; } My Account | Register | Help Flag as Inappropriate This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate?          Excessive Violence          Sexual Content          Political / Social Email this Article Email Address:

Matrix of ones

Article Id: WHEBN0000690246
Reproduction Date:

 Title: Matrix of ones Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

Matrix of ones

In mathematics, a matrix of ones or all-ones matrix is a matrix where every element is equal to one. Examples of standard notation are given below:

J_2=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix};\quad J_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix};\quad J_{2,5}=\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}.\quad

Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different matrix.

Properties

For an n×n matrix of ones J, the following properties hold:

• The trace of J is n, and the determinant is 1 if n is 1, or 0 otherwise.
• The rank of J is 1 and the eigenvalues are n (once) and 0 (n-1 times).
• J is positive semi-definite matrix. This follows from the previous property.
• J^k = n^{k-1} J, \mbox{ for } k=1,2,\ldots.\,
• The matrix \tfrac1n J is idempotent. This is a simple corollary of the above.
• \exp(J) = I + \frac{ e^n-1}{n} J, where exp(J) is the matrix exponential.
• J is the neutral element of the Hadamard product.
• If A is the adjacency matrix of a 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.