The original meaning of the mother is the uterus, the mother who controls the center and the place where life is born. Mathematically, a matrix refers to a two-dimensional data table arranged vertically and horizontally, which originated from a square matrix composed of coefficients and constants of an equation. This concept was first put forward by British mathematician Kelly in19th century. The concept of matrix also has many applications in production practice, such as the matrix diagram method for protecting personal accounts and the matrix card system (proposed in Shenzhen domain). The original meaning of "matrix" is also commonly used. For example, the analog device responsible for the switching control of the front-end video source and control line in the monitoring system is also called matrix.

The following is a 4 × 3 matrix:

The I-th and J-th columns of matrix A, or I and J bits, are usually recorded as A=7.

In c language, it is also expressed as A[j]. (It is worth noting that, unlike the general matrix algorithm, in C, "rows" and "columns" are counted from 0. )

In addition, A = (aij), which means that a [I, j] = aiji is common in all I and J in mathematical works.

Matrices constructed over general rings

It is given that a ring R, M(m, n, r) is the set of all m× n matrices in R arranged by elements. If m=n, it is usually denoted as M(n, r). These matrices can be added and multiplied (see below), so M(n, r) itself is a ring, which is isomorphic to the endomorphism ring of the left R module Rn.

If R is replaceable, then M(n, r) is an R- algebra with identity. Determinant can be defined by Leibniz formula: a matrix is invertible if and only if its determinant is invertible within R.

In Baidu Encyclopedia, unless otherwise specified, a matrix is mostly a real matrix or a virtual matrix.

partitioned matrix

Block matrix refers to a matrix that divides a large matrix into "matrices". For example, the following matrix

It can be divided into four 2×2 matrices, which can control various signals freely and display BSV LCD in multi-screen mode.

This method can be used to simplify operation, mathematical proof and some computer applications, such as VLSI chip design.

A symmetric matrix is symmetric with respect to its main diagonal (from upper left to lower right), that is, ai, j = aj, i.

Hermite matrix (or self-yoke matrix) is symmetrical with respect to its main diagonal in the form of complex yoke, that is, ai, j = a * j, i.

All elements of the Teplic matrix are opposite on any diagonal, that is, ai, j=ai+ 1, j+ 1.

All columns of a random matrix are probability vectors, which are used for Markov chains.

In addition, there are diagonal matrices, identity matrix matrices and banded matrices.

[1] The diagonal matrix has elements only on its main diagonal, and all elements in other positions are zero (that is, aij=

0 or i≠j). As shown in the figure, it is the diagonal matrix of nXn:

Similar to identity matrix, but the elements on the main diagonal are all 1, that is, A 1 = A2 =...= An = 1.

A strip matrix refers to a matrix with non-zero elements in the position parallel to the main diagonal, and all elements in other positions are zero.

English name Matrix(SAMND matrix). In mathematical terms, matrices are used to represent various related data such as statistical data. This definition well explains the mathematical and logical basis of the manufacturing world of matrix codes.

Nine Chapters Arithmetic, written at the end of the Western Han Dynasty and the beginning of the Eastern Han Dynasty, used the separation coefficient method to represent the linear equations and obtained its augmented matrix. In the process of elimination, one line is multiplied by a non-zero real number and one line is subtracted from another, which is equivalent to the elementary transformation of a matrix. However, the concept of matrix was not understood at that time. Although it is the same as the matrix now, it was only used as the standard representation and processing method of linear equations at that time.

The modern concept of matrix was gradually formed in19th century. German mathematician F. Gauss (1777~ 1855) takes all the coefficients of a linear transformation as a whole. 1844, German mathematician F. Eisenstein (1823 ~1852) discussed "transformation" (matrix) and its product. 1850, the English mathematician james joseph sylvester (18414-1897) first used the word matrix. 1858, the British mathematician A. Gehry (182 1~ 1895) published a research report on matrix theory. He first studied matrix as an independent mathematical object and published a series of articles on this subject, so he was regarded as the founder of matrix theory. He gave a series of commonly used definitions, such as equality of two matrices, zero matrix, identity matrix, sum of two matrices, quantitative product of a number and a matrix, product of two matrices, inverse of a matrix, transposed matrix and so on. Moreover, Gloria also noticed that the multiplication of matrices is combinable, but generally not commutative, and the m*n matrix can only be multiplied by the n*k matrix to the right. 1854, French mathematician C. Hermite (1822 ~1901) used the term "orthogonal matrix", but German mathematician F.G.Frohenius (1878) did not. In 1879, Ferrobenius introduced the concept of matrix rank.

At this point, the matrix system is basically established.