r - 外积如何在 R 中工作？

Question

作为尝试对循环进行矢量化的一部分，我偶然发现了 R 中的 outer(X, Y, FUN = "*", ...) 函数。

我试图了解如何逐步重现以下结果:

set.seed(1)
b = rnorm(3, 0, 1)
t = rnorm(5)

使用 outer() 和 FUN 参数作为 - 我得到以下输出:

> outer(t, b, "-")
           [,1]       [,2]        [,3]
[1,] -0.9134962 -1.7235934 -0.70432143
[2,] -0.3021132 -1.1122104 -0.09293842
[3,]  0.3317334 -0.4783638  0.54090817
[4,]  0.6206866 -0.1894105  0.82986144
[5,]  3.0311072  2.2210101  3.24028200

使用 outer() 和 FUN 参数作为 * 我得到:

> outer(t, b, "*")
             [,1]         [,2]         [,3]
[1,]  0.964707572 -0.282801545  1.286826317
[2,]  0.581704357 -0.170525137  0.775937183
[3,]  0.184628747 -0.054123443  0.246276838
[4,]  0.003612867 -0.001059103  0.004819215
[5,] -1.506404279  0.441598542 -2.009397175

我可以通过 t %*% t(b) 重现 outer(t, b, "*")，但我想不通如何为 outer(t, b, "-") 做这件事。

我对矩阵代数的了解相当有限，但我想试一试。你能帮帮我吗:

• 重现 FUN 设置为 -
的情况
• 阐明 FUN 的实际作用？

谢谢。

Answer 1

问题是从 stats.stackexchange 迁移而来的，原始答案包括数学方程式。您可以在下面找到原始文本，以及保留格式的图片。

图片(保留格式)

原文

The outer product of two vectors $x,y$ (which do not need to have the same dimension) is often written $x y^T$ or, with more details, $$ \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix} \cdot
\begin{pmatrix} y_1 & y_2 & \dots & y_m \end{pmatrix} $$ and the result is the $n \times m $ matrix $$ \begin{pmatrix} x_1 y_1 & x_1 y_2 & \dots x_1 y_m \ x_2 y_1 & x_2 y_2 & \dots x_2 y_m \ \vdots \ x_n y_1 & x_n y_2 & \dots & x_n y_m \end{pmatrix} $$ So, you can see, the result is an $n\times m$ matrix where element $i,j$ is given by $x_i \cdot y_j$. So this is outer product where the FUN is ordinary multiplication. In general, the result is the same, always an $n \times m$-matrix, where ordinary multiplication is replaced with an arbitrary two-place function $\text{FUN}(x,y)$, so if that function is ordinary minus, $-$ then the $i,j$ element becomes $x_i - y_j$, if FUN is power, $\text{FUN}(x,y) = x^y$ then the $i,j$ element becomes $x_i^{y_j}$, and so on.

This could even be used with non-numerical functions.