In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a matrix of ones or all-ones matrix is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
where every entry is equal to
one. Examples of standard notation are given below:
:
Some sources call the all-ones matrix the unit matrix, but that term may also refer to the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, a different 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 vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
s''.
Properties
For an matrix of ones ''J'', the following properties hold:
* The
trace of ''J'' equals ''n'', and the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
equals 0 for ''n'' ≥ 2, but equals 1 if ''n'' = 1.
* The
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
of ''J'' is
.
* The
minimal polynomial of ''J'' is
.
* The
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of ''J'' is 1 and the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s are ''n'' with
multiplicity
Multiplicity may refer to: In science and the humanities
* Multiplicity (mathematics), the number of times an element is repeated in a multiset
* Multiplicity (philosophy), a philosophical concept
* Multiplicity (psychology), having or using mult ...
1 and 0 with multiplicity .
*
for
[.]
* ''J'' is the
neutral element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the
Hadamard product.
When ''J'' is considered as a matrix over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, the following additional properties hold:
* ''J'' is
positive semi-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
.
*The matrix
is
idempotent
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
.
*The
matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
of ''J'' is
Applications
The all-ones matrix arises in the mathematical field of
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, particularly involving the application of algebraic methods to
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
. For example, if ''A'' is the
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
of an ''n''-vertex
undirected graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
''G'', and ''J'' is the all-ones matrix of the same dimension, then ''G'' is a
regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...
if and only if ''AJ'' = ''JA''.
[.] As a second example, the matrix appears in some linear-algebraic proofs of
Cayley's formula
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^.
The formula equivalently counts the number of spanning tr ...
, which gives the number of
spanning tree
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not ...
s of a
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...
, using the
matrix tree theorem
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time ...
.
See also
*
Zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed ...
, a matrix where all entries are zero
*
Single-entry matrix
In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as E_. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is
E_ = \beg ...
References
Matrices
1 (number)
{{Linear-algebra-stub