From Surf Wiki (app.surf) — the open knowledge base

Matrix of ones

Matrix with every entry equal to one

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

: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, 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 n × n 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 (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 n − 1.
J^k = n^{k-1} J for k = 1,2,\ldots .
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 \tfrac1n J is idempotent.
The matrix exponential of J is \exp(\mu J)=I+\frac{e^{\mu n}-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. 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 (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.{{citation

References

(2012). "Matrix Analysis". Cambridge University Press.
"Unit Matrix".
Stanley, Richard P.. (2013). "Algebraic Combinatorics: Walks, Trees, Tableaux, and More". Springer.
{{harvtxt. Stanley. 2013; {{harvtxt. Horn. Johnson
Timm, Neil H.. (2002). "Applied Multivariate Analysis". Springer.
Smith, Jonathan D. H.. (2011). "Introduction to Abstract Algebra". CRC Press.
Godsil, Chris. (1993). "Algebraic Combinatorics". CRC Press.

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

matrices-(mathematics) 1-(number)

Want to explore this topic further?

Ask Mako anything about Matrix of ones — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report