解决非平方矩阵中的线性方程组 [英] Solving a system of linear equations in a non-square matrix

问题描述

我有一个线性方程组，它组成一个我需要求解的- NxM 矩阵(即非平方)-或至少要尝试尝试为了表明该系统没有解决方案. (很有可能不会有解决方案)

I have a system of linear equations that make up an NxM matrix (i.e. Non-square) which I need to solve - or at least attempt to solve in order to show that there is no solution to the system. (more likely than not, there will be no solution)

据我了解，如果我的矩阵不是正方形(超出或未确定)，那么找不到精确解-我在想这个吗?有没有一种方法可以将我的矩阵转换为方矩阵，以便计算确定值，应用高斯消去法，克莱默法则等?

As I understand it, if my matrix is not square (over or under-determined), then no exact solution can be found - am I correct in thinking this? Is there a way to transform my matrix into a square matrix in order to calculate the determinate, apply Gaussian Elimination, Cramer's rule, etc?

也许值得一提的是，我的未知数的系数可能为零，因此在某些罕见的情况下，零列或零行是可能的.

It may be worth mentioning that the coefficients of my unknowns may be zero, so in certain, rare cases it would be possible to have a zero-column or zero-row.

推荐答案

矩阵是否为正方形不是决定解决方案空间的因素.相对于确定列的列数，是矩阵的秩(请参阅秩为零定理).通常，根据线性方程组的等级和无效关系，您可以具有零个，一个或无限个解.

Whether or not your matrix is square is not what determines the solution space. It is the rank of the matrix compared to the number of columns that determines that (see the rank-nullity theorem). In general you can have zero, one or an infinite number of solutions to a linear system of equations, depending on its rank and nullity relationship.

但是，要回答您的问题，您可以使用高斯消去法来找到矩阵的秩，如果这表明存在解，则可以找到特定的解x0和矩阵的空空间Null(A).然后，您可以将所有解决方案描述为x = x0 + xn，其中xn代表Null(A)的任何元素.例如，如果矩阵是满秩的，则其零空间将为空，而线性系统最多将具有一个解.如果其等级也等于行数，则您有一个唯一的解决方案.如果零空间的尺寸为一，那么您的解决方案将是一条穿过x0的线，该线上的任何点都满足线性方程.

To answer your question, however, you can use Gaussian elimination to find the rank of the matrix and, if this indicates that solutions exist, find a particular solution x0 and the nullspace Null(A) of the matrix. Then, you can describe all your solutions as x = x0 + xn, where xn represents any element of Null(A). For example, if a matrix is full rank its nullspace will be empty and the linear system will have at most one solution. If its rank is also equal to the number of rows, then you have one unique solution. If the nullspace is of dimension one, then your solution will be a line that passes through x0, any point on that line satisfying the linear equations.

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