The concept of the matrix and basic definitions
Such a concept as a matrix is used and appears in the sections of higher mathematics. Most often it is found in the system. So what is the basic concept of a matrix? To understand this, it is not necessary to be a mathematical genius.
A matrix is something that is a system of numbers mn, which in mathematics are written in the form of tables, in which m are the rows and n are the columns. At its intersection are elements. The size of the matrix depends on the number of rows and columns.
Initially, a triangular matrix was considered, but it is not convenient for solving complex equations, so a rectangular analogue is used in mathematics. Such a system is considered to be convenient for solving the problem.
Where did the matrix come from
If we talk about the first mention, the concept of the matrix came from China, or rather it was associated with "magic squares". These tables were considered amulets due to the numbers inscribed on them. At that time there was neither the very concept of the matrix, nor the ways to solve it.In some tribes she determined the degree of kinship so that people would find out whether they could marry or not.
But the concept itself was introduced in the early 19th century by James Silvestrom, who was an English mathematician. Various scientists worked on the theory of matrices, creating a complex of solutions for difficult algebraic problems.
Matrix and its main types
So, earlier the concept of a matrix was considered. Types of matrices are presented below:
- Lowercase It consists of one row, that is, in this type of matrix, the number of rows is exactly m = 1, but the number of its columns is arbitrary.
- Columnar. This variety consists of only one column, but can also contain several rows. For this kind of matrix has its own universal entry. It indicates that items that are in parentheses should be written as a column. On top of that, it saves not only space, but also time.
- Null matrix Such a form exists when all its elements are equal to 0. They denote this kind of matrix "O" This element also received its name due to the fact that in calculus it has similar functions with zero in number theory.
- SquareThis type of matrix is the most common, its difference lies in the fact that it has an equal number of columns and rows. When two elements are connected at different ends of the column, we get the main diagonal. In this regard, the diagonal table is divided into several systems:
a) Diagonal. This type of square matrix, which does not include the elements belonging to the main diagonal.
b) Single. In this type of diagonal consists of only one unit. With all this, each such matrix is scalar.
c) Scalar. This is the type where the main diagonal consists of the same number.
d) Arrow. This is a kind of square matrix in which almost all elements are zero, but the components that make up the first column, as well as those related to the diagonal and the first row, are not equal to it.
- Binary This type of table consists of zeros and ones.
- Pascal's matrix - in this kind of matrix all elements consist of numbers belonging to the triangle of the same name.
- Tridiagonal In this kind of matrix, nonzero elements are located on three diagonals, which include the main one and those above and below.
Having learned that it has different types, it is necessary to say about the following concept: the rank of a matrix. By this is meant the number of independent linear columns or the number of independent linear rows. Also, the rank of any matrix is the order of the minors, which is the highest, its determinant is non-zero.
Rank is indicated by rang A, but rank A. is also found. Both concepts are considered correct, so with different spellings this will not be considered an error. The main thing is not to forget that the rank of rows is always equal to the rank of columns.
Actions on the matrix
The matrix is a universal system for solving algebraic problems; various actions are performed on it. These include the multiplication of two matrices, addition, multiplication by its number and other matrices.
Performing numerous actions in algebra, a system of various matrices helps to find an easier way to solve the problems. It takes less time and helps not to get lost in a lot of actions.
Matrix-related operations are used not only in mathematics, but also in physics and chemistry.