Inverse matrix formula of 1. square matrix;

If the square matrix A is invertible (that is, its determinant is not zero), the inverse matrix of the square matrix A is marked as A- 1, which satisfies the following formula: a× a-1 = a-1× a = i. Where I is identity matrix, diagonal elements are1,and other elements are 0.

2. Calculation formula of determinant of square matrix:

Determinant is the scalar value of a square matrix, which is represented by det(A), where a is an n×n square matrix. The formula for calculating the determinant of a square matrix is as follows: det (a) = a11+a12c12+...+a1NC1n. Where aij represents the element of row I and column J of matrix A, and Cij represents the algebraic cofactor of the element.

3. The eigenvalues and eigenvectors of the square matrix:

For a square matrix A, if there is a number λ and a non-zero vector V, and Av=λv is satisfied, λ is called the eigenvalue of A, and V is the eigenvector corresponding to the eigenvalue λ.

4. The characteristic polynomial of the square matrix:

For square matrix A, the characteristic polynomial P(λ) is defined as P(λ)=det(A-λI), where i is identity matrix. The characteristic polynomial can be used to solve the eigenvalue of a square matrix.

5. Diagonalization of square matrix;

If a square matrix A can be expressed as a = PDP- 1, where D is diagonal matrix and P is invertible matrix, then A is said to be diagonalizable. Diagonalization can simplify the operation and analysis of matrix, and whether a square matrix can be diagonalized can be judged by solving eigenvalue and eigenvector.

Knowledge expansion

Square matrix, also known as matrix or rectangular array, is a two-dimensional array composed of m rows and n columns of elements. It is a basic concept in linear algebra and widely used in mathematics, physics, computer science and other fields.

A square matrix with m rows and n columns can be expressed as an m×n rectangular table, in which each element can be determined by the coordinates of rows and columns. Each element can be any type of data, such as numbers, symbols, letters, functions, etc.

Square matrices have many important properties and operation rules. For example, you can add and subtract two square matrices, or you can multiply a square matrix with a scalar (that is, a constant). In addition, the square matrix also supports operations such as transposition, inverse matrix, determinant and eigenvalue.