线性方程解法（齐次，非齐次）

本文主要是介绍线性方程解法（齐次，非齐次），希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！

转载地址：https://en.wikipedia.org/wiki/System_of_linear_equations#Solving_a_linear_system

由于很多地方需要求解齐次，非齐次线性方程，把线代，高数书毕业了都扔了，汗，简单总结一下。

实际中都是用Eigen 库，调用API解线性方程组，得注意每个函数的适用条件。

$A{\mathbf {x}}={\mathbf {b}}$

where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.

$A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad {\mathbf {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad {\mathbf {b}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}$

其中：A的行向量之间线性无关，这样每一行带来的才是关于参数新的约束信息。

In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns:

Usually, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system.
Usually, a system with the same number of equations and unknowns has a single unique solution.
Usually, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.（在实际中，我们往往用线性最小二乘的方法求X，使得|| b-AX || 最小）

$A{\textbf {x}}={\textbf {0}}$ 的解只有两种：

零解和非零解(无数多个)。

A假设为： m*n , r(A)=r

当 r=n, 只有零解；

当 r<n , n-r个基础解系，即有非零解；

$A{\textbf {x}}={\textbf {0}}$ 的解： SVD

Solving homogeneous linear equations[edit]

A set of homogeneous linear equations can be written as Ax = 0 for a matrix A and vector x. A typical situation is that A is known and a non-zero x is to be determined which satisfies the equation. Such an x belongs to A's null space and is sometimes called a (right) null vector of A.

解： The vector x can be characterized as a right-singular vector corresponding to a singular value of A that is zero.

This observation means that if A is a square matrix and has no vanishing singular value, the equation has no non-zero x as a solution. It also means that if there are several vanishing(零值)singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero x satisfying x∗A = 0, with x∗ denoting the conjugate transpose of x, is called a left null vector of A.

AX=b的解

If the equation system is expressed in the matrix form $A{\mathbf {x}}={\mathbf {b}}$ , the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution given by

${\mathbf {x}}=A^{-1}{\mathbf {b}}$