Understanding matrix

What is a matrix

A matrix is ​​a set of complex or real numbers arranged in a rectangular array. A matrix with real numbers is called a real matrix, and a matrix with elements is a complex matrix. A matrix with the number of rows and columns equal to n is called an n-th order matrix or an n-th order square matrix.

A number table of m rows and n columns arranged by m × n numbers aij is called a matrix of m rows and n columns, which is referred to as an m × n matrix. Referred to as:

Matrix representation

 

Relationship of scalar, vector, matrix, tensor

These 4 concepts are constantly rising in dimensionality. It is easier to understand the metaphorical explanation with the concept of point-line polyhedron:

  • Point - scalar (Scalar
  • Line - vectorvector
  • Face-matrix
  • Body - tensor

Relationship of scalar, vector, matrix, tensor

Interested parties can learn more about the following:

'Understanding scalars in one article"

'Understanding Vectors in One Article"

'Understanding matrices in one article"

'Understand tensor in one article"

 

Baidu Encyclopedia and Wikipedia

Baidu Encyclopedia version

In mathematics, a matrix is ​​a set of complex or real numbers arranged in a rectangular matrix, starting from the square matrix of coefficients and constants of the system of equations. This concept was first proposed by 19 century British mathematician Kelly.

Matrices are common tools in advanced algebra and are also commonly used in applied mathematics such as statistical analysis. In physics, matrices are used in circuit science, mechanics, optics, and quantum physics; in computer science, 3D animation requires matrix. The operation of the matrix is ​​an important issue in the field of numerical analysis. Decomposing a matrix into a simple matrix simplifies the operation of the matrix in both theoretical and practical applications. For some widely used and form-specific matrices, such as sparse matrices and quasi-diagonal matrices, there are specific fast arithmetic algorithms. For the development and application of matrix related theory, please refer to matrix theory. In the fields of astrophysics, quantum mechanics, etc., there are also infinite dimensional matrices, which is a generalization of matrices.

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Wikipedia version

In mathematics, a matrix is ​​a rectangular array of numbers, symbols, or expressions arranged in rows and columns. For example, the size of the matrix below is 2×3 (read “two by three”) because there are two rows and three columns:

If they have the same size (each matrix has the same number of rows and the same number of columns as the other), you can add or subtract two matrices element by element (see coincidence matrix). However, the rule of matrix multiplication is that two matrices can be multiplied only when the number of columns in the first column is equal to the number of rows in the second column (ie, the internal dimensions are the same, n is (m × n)) – Matrix Multiply the (n × p) matrix to get (m × p)-matrix. Conversely there is no product, the first one implies that matrix multiplication is not commutative. Any matrix can be multiplied element by element by the scalar in its related field. In each item m×n matrix A, it is often expressed as an I, J, in which I and J usually change from 1 to m and N are called its elements or entries respectively.

In order to conveniently represent the elements of the result of the matrix operation, the index of the element is usually appended to the matrix expression with parentheses or parentheses; for example: (AB)i, j refers to the element of the matrix product. In the context of abstract index representation, this ambiguity also refers to the entire matrix product.

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