matlab矩阵中的 */.*

Posted 2022-06-09 流浪若相惜

说实话，我知道今天才会意识到其实这些运算困扰了我不止一次两次，每次都会去查Help文档，真的是困惑的不行不行，因此，打算今天彻底的将其捋一遍。

文章目录

• • • • 点乘（·*）与星乘 *
      • 右除（/）与左除（\\）

点乘（·*）与星乘 *

点乘（·*）：矩阵A和B对应项相乘
类型：(1) A：m×n; B: m×n (2) A:m×n; B:1×n （列要一致）（3）A：m×n；B：m×1 （行一致），且后两种情况为行或者列向量时才可成立，否则需要两个矩阵具有相同的行列维度
例：

x3 =
     2     3
     3     4
y3 =
     2     1
x3.*y3
ans =
     4     3
     6     4
y4 =
     2
     1
x3.*y4
ans =
     4     6
     3     4

星乘（*）：矩阵A的列项数与矩阵B的行项数一致
类型：A：m×n； B：n×l
例：

x3 =
     2     3
     3     4
y4 =
     2
     1
x3*y4
ans =
     7
    10

右除（/）与左除（\\）

左除：Solve systems of linear equations Ax = B for x (mldivide)
矩阵左除有以下几种情况：

x = A\\B solves the system of linear equations A*x = B. The matrices A and B must have the same number of rows. MATLAB® displays a warning message if A is badly scaled or nearly singular, but performs the calculation regardless.
If A is a scalar, then A\\B is equivalent to A.\\B.
If A is a square n-by-n matrix and B is a matrix with n rows, then x = A\\B is a solution to the equation A*x = B, if it exists.
If A is a rectangular m-by-n matrix with m ~= n, and B is a matrix with m rows, then A\\B returns a least-squares solution to the system of equations A*x= B.

也就是说若A是矩阵，则矩阵B必须要与A有相同的行数
例：

A = [1 2 0; 0 4 3];
b = [8; 18];
x = A\\b
x = 

         0
    4.0000
    0.6667

这个函数实则是求解线性方程的，即，
$$\begin{array}{l}3x1+2x2+1=6\\ 3x2+1=7\end{array}$$
这里A=[3 2 1;0 3 1] ;X=[x1;x2;1]，B=[6;7],因此X=A\\B
那么对于回归分析来说，则系数矩阵代表的是X，变量则是A（样本值），因此X=A\\B,求得的是系数矩阵
右除：Solve systems of linear equations xA = B for x （mrdivide）
矩阵右除有三种情况：

x = B/A solves the system of linear equations x*A = B for x. The matrices A and B must contain the same number of columns. MATLAB® displays a warning message if A is badly scaled or nearly singular, but performs the calculation regardless.
If A is a scalar, then B/A is equivalent to B./A.
If A is a square n-by-n matrix and B is a matrix with n columns, then x = B/A is a solution to the equation x*A = B, if it exists.
If A is a rectangular m-by-n matrix with m ~= n, and B is a matrix with n columns, then x = B/A returns a least-squares solution of the system of equations x*A = B

右除当A为矩阵时，要求B为与A同列项数的矩阵
例：

A = [1 1 3; 2 0 4; -1 6 -1];
B = [2 19 8];
x = B/A
x = 

    1.0000    2.0000    3.0000