# 一、什么是旋转矩阵？

[En]

First of all, I need to take a moment to talk about the meaning of the rotation matrix. You must know that the rotation matrix has two meanings: * coordinate transformation * and * rotation vector . In order for you to understand, I will give the simplest example.*

## 1.坐标变换

[En]

The transformation relation is derived from the geometric relation, regardless of the details, let’s see what the above formula looks like in matrix form:

[En]

What causes it? It’s easiest to understand from an algebraic point of view. Please don’t bother to follow me step by step:

[En]

It can be seen that the column vector of the rotation matrix written according to the geometric relationship happens to be the coordinate of the coordinate axis of the rotating coordinate system (red) in the coordinate system (black) before rotation, or the direction cosine.

## ; 2.向量旋转

[En]

In addition to the function of coordinate transformation, another function of the rotation matrix is that it can rotate the vector, which we can still consider from an algebraic point of view.

[En]

I thought a lot of words to describe this operation, and later found that the drawing is still intuitive, please read on:

[En]

So far, I think you have understood the two functions of the rotation matrix, one is coordinate transformation, the other is vector rotation, the same rotation matrix, the same formula, it depends on how you explain it, this point is very important, we must be clear.

[En]

The top is the coordinate transformation, the bottom is the vector rotation, the rotation matrix must be the same, what if the vector in the blue box on my right is the same? So is the value on the left the same? So what exactly is the physical meaning of the left and right values? It depends on how you explain it, whether you interpret it as a vector rotation or as a coordinate transformation.

# 二、矩阵的左右乘

## ; 2.旋转矩阵右乘

[En]

If the logic doesn’t make sense here, just think about it for a while, (brain repair how to rotate, this time around its own coordinate system).

# 总结

Original: https://blog.csdn.net/weixin_45632220/article/details/117735223
Author: 默以成之
Title: 旋转矩阵及左右乘的意义，看这一篇就够了

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