# Matrix unit

Matrix unit

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 semigroups.

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 numbers (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:

• 0 x = x 0 = 0 for any matrix unit x;
• (r, c) (s, d) = (r, d) if c = s, and 0 if cs.

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+)

(2, 3) (3, 2) (2, 1) (1, 4) = (2, 4)

represents the abstract matrix multiplication

$\begin{bmatrix} \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \end{bmatrix} \begin{bmatrix} \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ \end{bmatrix} \begin{bmatrix} \cdot \\ \cdot \\ \end{bmatrix} \begin{bmatrix} \cdot & \cdot & \cdot & \cdot \\ \end{bmatrix} = \begin{bmatrix} \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \end{bmatrix}$.

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 of the bicyclic semigroup, meaning that it summarises how composition for members of that class interacts with the structure of the semigroup's principal ideals.

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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• matrix unit — matricinis įtaisas statusas T sritis radioelektronika atitikmenys: angl. matrix unit vok. Matrixeinheit, f rus. матричное устройство, n pranc. dispositif matriciel, f …   Radioelektronikos terminų žodynas

• unit matrix — vienetinė matrica statusas T sritis fizika atitikmenys: angl. identity matrix; unit matrix vok. Einheitsmatrix, f rus. единичная матрица, f pranc. matrice unité, f; matrice unitaire, f …   Fizikos terminų žodynas

• Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

• Matrix mechanics — Quantum mechanics Uncertainty principle …   Wikipedia

• Unit fraction — A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/ n . Examples are 1/1, 1/2, 1/3, 1/42 etc.… …   Wikipedia

• Unit vector — In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a hat , like this: (pronounced i hat ). In Euclidean space …   Wikipedia

• matrix — /may triks, ma /, n., pl. matrices /may tri seez , ma /, matrixes. 1. something that constitutes the place or point from which something else originates, takes form, or develops: The Greco Roman world was the matrix for Western civilization. 2.… …   Universalium

• Matrix of ones — In mathematics, a matrix of ones is a matrix where every element is equal to one. Examples of standard notation are given below: In special contexts, the term unit matrix is used as a synonym for matrix of ones [1] This is done whenever it is… …   Wikipedia

• Unit (ring theory) — In mathematics, a unit in a (unital) ring R is an invertible element of R , i.e. an element u such that there is a v in R with : uv = vu = 1 R , where 1 R is the multiplicative identity element.That is, u is an invertible element of the… …   Wikipedia

• Matrix determinant lemma — In mathematics, in particular linear algebra, the matrix determinant lemma[1][2] computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT. Contents 1 Statemen …   Wikipedia