Matrix unit
Encyclopedia
In mathematics
, a matrix unit is an idealisation of the concept of a matrix
, with a focus on the algebraic properties of matrix multiplication
. The topic is comparatively obscure within linear algebra
, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra
, especially the theory of semigroup
s.
Despite the name, matrix units are not the same as unit matrices
or unitary matrices.
Two matrices can be multiplied when the number of columns in one is the same as the number of rows in the other; otherwise, they are incompatible. The idea behind matrix units is to look at this fact in isolation: a matrix unit is a matrix with dimensions, but with the entries scooped out.
Let I be a nonempty set, to be used for counting the matrix rows and columns. There is no requirement for it to be finite; indeed, standard matrix algebra would use the set of natural number
s (not including zero) N+. A matrix unit is either an ordered pair
(r, c), with r and c elements of I, or it is a special "zero" object, written as "0". Multiplication is defined as follows:
The 0 element can be seen as an "error symbol" for when multiplication fails; the first rule implies that errors propagate through an entire product containing a single incompatible combination.
For example, the product (with I = N+) (3, 2) (2, 1) (1, 4) = (2, 4)
represents the abstract matrix multiplication
.
Another notation for (r, c) is Ar c, following the convention for naming a single entry of a matrix. (Different letters are used in the "A" position to refer to matrix units on a different base set.) The composition rule may be expressed using the Kronecker delta as
With these rules, (I × I) ∪ {0} is a semigroup with zero. Its construction is analogous to that for other important semigroups, such as rectangular bands and Rees matrix semigroups. It also arises as the trace of the unique D-class
of the bicyclic semigroup
, meaning that it summarises how composition for members of that class interacts with the structure of the semigroup's principal ideal
s.
A semigroup of matrix units is 0-simple, because any two nonzero elements generate the same two-sided ideal (the entire semigroup), and the semigroup is non-null. The elements (r, c) and (s, d) are D-related via
as any pairs are R-related if they have the same first coordinate and L-related if they have the same second coordinate. All H-classes are singletons. The idempotents are the "square" matrix units (a, a) for a in I, together with 0.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a matrix unit is an idealisation of the concept of a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, with a focus on the algebraic properties of matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
. The topic is comparatively obscure within linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, especially the theory of semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
s.
Despite the name, matrix units are not the same as unit matrices
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
or unitary matrices.
Two matrices can be multiplied when the number of columns in one is the same as the number of rows in the other; otherwise, they are incompatible. The idea behind matrix units is to look at this fact in isolation: a matrix unit is a matrix with dimensions, but with the entries scooped out.
Let I be a nonempty set, to be used for counting the matrix rows and columns. There is no requirement for it to be finite; indeed, standard matrix algebra would use the set of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (not including zero) N+. A matrix unit is either an ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
(r, c), with r and c elements of I, or it is a special "zero" object, written as "0". Multiplication is defined as follows:
- 0 x = x 0 = 0 for any matrix unit x;
- (r, c) (s, d) = (r, d) if c = s, and 0 if c ≠ s.
The 0 element can be seen as an "error symbol" for when multiplication fails; the first rule implies that errors propagate through an entire product containing a single incompatible combination.
For example, the product (with I = N+) (3, 2) (2, 1) (1, 4) = (2, 4)
represents the abstract matrix multiplication
.Another notation for (r, c) is Ar c, following the convention for naming a single entry of a matrix. (Different letters are used in the "A" position to refer to matrix units on a different base set.) The composition rule may be expressed using the Kronecker delta as
- Xr c Xs d = δc s Xr d.
With these rules, (I × I) ∪ {0} is a semigroup with zero. Its construction is analogous to that for other important semigroups, such as rectangular bands and Rees matrix semigroups. It also arises as the trace of the unique D-class
Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
of the bicyclic semigroup
Bicyclic semigroup
In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H...
, meaning that it summarises how composition for members of that class interacts with the structure of the semigroup's principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
s.
A semigroup of matrix units is 0-simple, because any two nonzero elements generate the same two-sided ideal (the entire semigroup), and the semigroup is non-null. The elements (r, c) and (s, d) are D-related via
- (r, c) R (r, d) L (s, d),
as any pairs are R-related if they have the same first coordinate and L-related if they have the same second coordinate. All H-classes are singletons. The idempotents are the "square" matrix units (a, a) for a in I, together with 0.