Matlab矢量化:如何避免这种“为".环形? [英] Matlab Vectorization : How to avoid this "for" loop?

问题描述

我有以下矩阵:

X=1 2 3  
Y=4 5 6  

A=1 2 3  
  4 5 6  
  7 8 9  

我想做

for each (i,j) in A  
  v = A(i,j)*X - Y
  B(i,j) = v * v'

即A的每个元素都乘以向量X，然后所得向量从其自身减去Y，最后我们取该向量的内积以得出单个数字.
可以不用for循环吗?

i.e. each element of A is multiplied by vector X, then resultant vector subtracts Y from itself and finally we take inner product of that vector to bring a single number.
Can it be done without for loop ?

推荐答案

在Matlab中经常忘记的一件事:运算符'采用转置的共轭(.'是普通的转置).换句话说，A' == conj(trans(A))，而A.' == trans(A)，如果A是复数矩阵，则会有所不同.

One thing often forgotten in Matlab: The operator ' takes the conjugate transposed (.' is the ordinary transposed). In other words, A' == conj(trans(A)), whereas A.' == trans(A), which makes a difference if A is a complex matrix.

好的，让我们将一些数学应用于方程式.我们有

Ok, let's apply some mathematics to your equations. We have

v = A(i,j)*X - Y
B(i,j) = v * v'
       = (A(i,j)*X - Y) * (A(i,j)*X - Y)'
       = A(i,j)*X * conj(A(i,j))*X' - Y * conj(A(i,j))*X' 
         - A(i,j)*X * Y' + Y * Y'
       = A(i,j)*conj(A(i,j)) * X*X' - conj(A(i,j)) * Y*X' - A(i,j) * X*Y' + Y*Y'

第一个结果将是

B = A.*conj(A) * (X*X') - conj(A) * (Y*X') - A * (X*Y') + Y*Y'

对于实矩阵/向量，一个具有身份

X*Y' == Y*X'
A == conj(A)

这意味着您可以将表达式简化为

which means, you can reduce the expression to

B = A.*A * (X*X') - 2*A * (X*Y') + Y*Y'
  = A.^2 * (X*X') - 2*A * (X*Y') + Y*Y'

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