The formula of multiplication and distribution law of matrix and number is λ (a+b) = λ a+λ b.
The most important method of matrix multiplication is general matrix multiplication, which makes sense only when the number of columns in the first matrix is the same as the number of rows in the second matrix.
In mathematics, a matrix is a group of complex numbers or real numbers arranged in a rectangular array, which originated from a square matrix composed of coefficients and constants of equations. This concept was first put forward by British mathematician Kelly in19th century. Senchai powder
Proof of matrix multiplication distribution law
Let A=(aij), B=(bij) and C=(cij), then
A(B+C)=(∑a[ip](b[pj]+c[pj]) (the numbers in the outer brackets of the giant chain represent the elements at I and J positions of a (b+c)).
=(∑(a[IP]b[pj]+a[IP]c[pj])(∑ only sums P, and the table in [] shows the subscript of mega chain) Sen guess scatter.
= (∑ This is a[ip]b[pj])+(∑a[ip]c[pj])
=AB+AC
use
An important use of matrix is to solve linear equations. The coefficients of unknown quantities in linear equations can be arranged into a matrix in megabytes, which is called augmented matrix when constant terms are added. Another important application is to express linear transformation, that is, the generalization of linear functions such as f(x)4x.
After the basis is set, a vector V can be expressed as a matrix of m× 1, and the linear transformation F can be expressed as a matrix of m, so that the vector f(v) obtained after the transformation can be expressed as Av, and the eigenvalues and eigenvectors of the matrix can reveal the deep features of the linear transformation.
Extended data
The most important method of matrix multiplication is generalized matrix product. It only makes sense when the number of columns in the first matrix is the same as the number of rows in the second matrix. When we generally refer to matrix product, we mean general matrix product. The matrix of M×N is a number matrix with M×N numbers arranged in m rows and n columns.
Because it compactly concentrates a large amount of data together, sometimes it can simply represent some complex models, such as power system network model.