Matlab右除法结果疑惑及Python等效实现咨询

Hey there! Let’s unpack MATLAB’s right division operator and why you might be seeing x * sum(m) ≠ m—plus how to replicate this behavior in Python.

What’s Happening in MATLAB? #

First, forget thinking of / as a simple "divide" operation. In MATLAB, A / B is defined as solving the linear equation x * B = A. The result depends on the dimensions and rank of B:

• If B is a square, invertible matrix, A/B is exactly A * inv(B), and x*B will perfectly equal A.
• If B is non-square (like your sum(m) being a row vector when m is a matrix) or singular, MATLAB returns the least-squares solution. This means it finds the x that minimizes the squared error norm(x*B - A)—there’s no exact solution to x*B = A, so this is the closest possible fit. That’s why you’re seeing x * sum(m) ≠ m!

Example in MATLAB #

Let’s use a concrete matrix to see this:

m = [1 2 3; 4 5 6];
s = sum(m); % s = [5 7 9] (a 1x3 row vector)
x = m / s;  % x is a 2x1 vector

Calculating x*s gives a 2x3 matrix that’s close to m, but not identical—it’s the best fit possible via least squares.

Replicating This in Python #

Python’s NumPy library has tools to match MATLAB’s right division behavior, depending on whether you need an exact or least-squares solution.

1. Least-Squares Solution (For Non-Square/Singular B) #

Use numpy.linalg.lstsq to directly solve the least-squares problem. We need to adjust dimensions slightly to match the equation format:

import numpy as np

m = np.array([[1, 2, 3], [4, 5, 6]])
s = m.sum(axis=0)  # Same as MATLAB's sum(m), gives [5,7,9]

# Solve x*s = m → equivalent to s.T * x.T = m.T
x_lstsq = np.linalg.lstsq(s.reshape(1, -1), m.T, rcond=None)[0].T

2. Using Pseudoinverse #

MATLAB’s right division for non-square cases under the hood uses the Moore-Penrose pseudoinverse. You can replicate this with numpy.linalg.pinv:

x_pinv = m @ np.linalg.pinv(s.reshape(1, -1))

This will give the exact same result as lstsq (and MATLAB’s / operator).

3. Exact Division (When sum(m) is a Scalar) #

If m is a vector, sum(m) is a scalar, and MATLAB’s m/sum(m) is just element-wise division. In Python, this is straightforward:

m = np.array([1,2,3])
s = m.sum()
x = m / s  # x*s will equal m exactly

内容的提问来源于stack exchange，提问作者emma qqq