Invariant Points and Lines in Matrix Transformations | 矩阵变换中的不变点与不变线

📚 Invariant Points and Lines in Matrix Transformations | 矩阵变换中的不变点与不变线

When a linear transformation is applied to the plane, most points move. A small number behave in a special way: they either stay exactly where they are, or they slide along a line that the transformation maps onto itself. These are the invariant points and invariant lines of the transformation, and they are one of the most reliable sources of marks in AQA A-Level Further Mathematics matrix questions.

当线性变换作用于平面时，绝大多数点都会移动。但有少数点很特别：它们要么原地不动，要么沿着一条被变换映回自身的直线滑动。这就是变换的不变点与不变线，也是 AQA A-Level 进阶数学矩阵题中最稳定的得分点之一。

1. What Are Invariant Points? | 什么是不变点

An invariant point of a transformation is a point that the transformation does not move at all. If the transformation is represented by the 2 × 2 matrix M and the position vector of the point is x, the defining condition is simply Mx = x.

变换的不变点，就是在变换下完全不会移动的点。若该变换由 2 × 2 矩阵 M 表示，点的位置向量为 x，定义条件就是 Mx = x。

The origin is always an invariant point of any matrix transformation, because M0 = 0 for every matrix M. The real question in an exam is whether any other points are also fixed.

原点永远是任何矩阵变换的不变点，因为对任意矩阵 M 都有 M0 = 0。考试中真正要问的是：除原点之外，是否还有别的点也不动。

Note the key word: an invariant point is fixed as a point. It does not merely land somewhere on the same line — it lands on itself.

注意关键词：不变点作为「点」是不动的，它不只是落在同一条直线上，而是落回它自己。

2. Finding Invariant Points Algebraically | 用代数方法求不变点

Rewrite Mx = x as (M − I)x = 0, where I is the 2 × 2 identity matrix. This is a pair of simultaneous homogeneous linear equations in x and y.

把 Mx = x 改写成 (M − I)x = 0，其中 I 是 2 × 2 单位矩阵。这是一个关于 x 与 y 的齐次二元一次方程组。

Because x = 0 is always a solution, a non-trivial solution exists only when the coefficient matrix M − I is singular. That gives the single condition you should remember:

由于 x = 0 永远是一个解，只有系数矩阵 M − I 奇异时才会出现非平凡解。由此得到唯一需要牢记的条件：

Mx = x ⇔ (M − I)x