Scalars, vectors, arrays, and matrices

Scalar （ Number ）

        Physical definition ： Only in size , A quantity without direction .（ Keep the same under coordinate transformation ）
        Corresponding physical examples ： quality 、 density 、 temperature 、 work 、 Energy and so on are only numerical values , A physical quantity without direction

From a dimensional point of view , Scalars can be considered one-dimensional , Similar particle , It can only be used to describe the quantity of something

vector （ vector ）

        Physical definition ： A quantity with size and direction .（ Keep the same under coordinate transformation ）
        Corresponding physical examples ： Displacement 、 Force, etc , And the physical quantity with direction

        There are generally two ways to represent a vector
       1. Coordinate representation ： Define a coordinate system corresponding to the dimension , Move the starting coordinates of the vector to the origin , At this point, the vector endpoint coordinates can represent the vector .（ It is formed by the unit vector corresponding to the coordinate axis ）
       2. A directed line segment represents ： Express by module and corresponding unit vector . Modulus is the length of the vector , The direction of the corresponding unit vector is consistent with the direction of the vector , And the length is 1 Vector .（ Of course, unit vectors are generally expressed by coordinates ）

        in summary , A vector is usually represented by multiple numbers （ Or say , describe ）, But that doesn't mean that a vector consists of scalars .

From a dimensional point of view , A vector is two-dimensional and above , A two-dimensional vector is also called a plane vector

matrix

        Mathematical concepts ： A set of values arranged in a rectangular array , A matrix generally refers to a two-dimensional matrix .

        Every element of a matrix is a scalar , Every line / Column （>1） Can represent a vector , namely n A matrix formed by a row of elements is a matrix n The row vector of dimension , A matrix formed in a row is a matrix n The column vector of dimension ,

        Matrix calculation method and scalar 、 Different vectors , Its special representation is convenient for calculating some related data . Matrix in statistical analysis 、 Circuit science 、 It has applications in optics and quantum physics .

From a dimensional point of view ,n Dimension vector regardless of dimension n How big? , It's just a thread , But a matrix is at least a face （ A two-dimensional ）, Cube （ The three dimensional ）,……

Array

        And the scalar above 、 vector 、 The difference between matrices is , Array is a programming language concept , A collection of mathematical concepts , Composed of elements . Arrays are also called ordered sequences of elements , The corresponding element can be accessed by subscript , The elements are generally of the same type .

        An array is almost equivalent to a matrix , And downward compatible with vector and scalar , The array elements themselves are scalars ; A one-dimensional array formed by combining array elements in one-dimensional way is a vector （ Similitude matrix ）（ Be careful ： One dimensional arrays can represent n Dimension vector ）; The two-dimensional array formed by combining the array elements in rows and columns is a two-dimensional matrix （ Be careful , Matrices have special operations , For example, the determinant value , Rank rank , But a two-dimensional array is usually only used to store data , except MATLAB）

        Arrays and matrices can be thought of as having the same form , The former is a collection of data in a programming language , The latter is mathematical language （ linear algebra ） A collection of data in . Arrays in programming languages have different dimensions , But it is generally only used to store the corresponding data and not to calculate with the entire array （ except MATLAB！！！）. The matrix in mathematical language also has multiple dimensions , The stored data usually has some close connection , For example, the coefficients of polynomial equations , And there will be various operations between matrices .

Personally think that , A matrix is a set of numerical values that mathematically describe a square array arrangement , Array is a way of data storage in programming language , Array can be used as a data storage method to imitate matrix , Of course , Most programming languages do not mimic the corresponding calculations .