numpy的基数变化 [英] Change of basis in numpy

问题描述

给出numpy中的两个基本矩阵 basis_old 和 basis_new ，库中某处有一个函数来获取转换矩阵以转换矢量 vec basis_old 中的code>到在 basis_new 中的表示形式?

Given two basis matrices basis_old and basis_new in numpy, is there a function somewhere in the library to get the transformation matrix to convert a vector vec in basis_old to its representation in basis_new?

例如，如果我在标准基础上有一个向量 vec = [1,2,3] [1,0,0]，[0,1,0]，[0,0,1] ，如何将其转换为另一个基础，例如

For example, if I have a vector vec = [1,2,3] in the standard basis [1,0,0], [0,1,0], [0,0,1], how to I convert it to another basis, say,

e1 = [1 0 0]
e2 = [0 0 1]
e3 = [0 1 0]
basis_new = np.array([e1, e2, e3])

# I want something like this
vec_new = np.linalg.change_of_basis(vec_old, basis_old, basis_new)

# Or this:
transformation_matrix = np.linalg.basis_change(basis_old, basis_new)

将base_new更改为线性独立

changed basis_new to be linearly independent

推荐答案

记住将向量w1，w2，w3作为R3的基础的含义.

Remember what it means for a set of vectors w1, w2, w3 to be a basis of R3.

1. w必须为 线性独立 .这意味着x1 w1 + x2 w2 + x3 w3 = 0的唯一解决方案应该是x1 = x2 = x3 =0.但是在您的情况下，您可以验证x1 = 1，x2 = -2，x3 = 1是另一种解决方案.所以你的 basis_new 无效.

矩阵W = [w1，w2，w3]必须是可逆的.

The matrix W = [w1, w2, w3] must be invertible.

对于R3中的每个向量，必须有一种唯一方式将其写成w的线性组合.

For every vector in R3 there must be a unique way to write it as a linear combination of w's.

一旦您满足了这些要求，便可以通过简单的矩阵乘法来计算新坐标.假设您要将向量v表示为v = c1 w1 + c2 w2 + c3 w3.要以矩阵形式写，v = W c.要获得c，您要做的就是将两边都乘以W的倒数.

Once you have nailed these requirements for a basis, then you can compute the new coordinates by a simple matrix multiplication. Suppose you want to express vector v as v = c1 w1 + c2 w2 + c3 w3. To write this in matrix form, v = W c. To get c, all you have to do is to multiply both side by the inverse of W.

c = W ^ {-1} v

c = W^{-1} v

在numpy中，您将其写为

In numpy, you would write it as,

vec_new = np.linalg.inv(np.array([w1, w2, w3])).dot(vec_old)

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