Invariant Points and Lines | 不变点与不变线

📚 Invariant Points and Lines | 不变点与不变线

In linear algebra and A-level matrix transformations, an invariant point is a point that is mapped to itself by a transformation, while an invariant line is a line whose points are all mapped to points on the same line. Understanding the difference between these two ideas is essential for solving transformation questions accurately.

在线性代数和 A-level 矩阵变换中，不变点是指被变换映射到自身的点；不变线是指线上所有点都被映射到同一条直线上的直线。准确理解这两个概念的区别是正确解决变换题目的关键。

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1. What is an invariant point? | 什么是不动点

A transformation T sends a point P with position vector p = (x, y)ᵀ if p′ = p. For a 2×2 matrix M, this means that the vector equation M p = p holds exactly.

一个变换 T 将位置向量为 p = (x, y)ᵀ 的点 P 映射到向量为 p′ = M p 的像点 P′。如果 p′ = p，则点 P 是不动点。对于 2×2 矩阵 M，这意味着向量方程 M p = p 精确成立。

The origin is always invariant under any linear transformation because M times the zero vector is always the zero vector. Non-origin invariant points exist only when the transformation has an eigenvalue equal to 1, or when the matrix is designed with a fixed line.

任何线性变换下，原点总是不动点，因为 M 乘以零向量始终是零向量。非原点的不动点只有在变换具有等于 1 的特征值，或矩阵具有固定直线时才存在。

M p = p

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2. Finding invariant points from a matrix transformation | 从矩阵变换求不变点

Write p = (x, y)ᵀ + by = x and y′ = cx + dy = y. Rearranging these produces two linear equations in x and y.

记 p = (x, y)ᵀ。如果 M = (a b; c d)，则条件 M p = p 给出联立方程组 x′ = ax + by = x 和 y′ = cx + dy = y。整理后可得到两个关于 x 和 y 的线性方程。

(a − 1)x + b y = 0, c x + (d − 1)y = 0

e origin is the only invariant point. If the equations reduce to a single line, then every point on that line is invariant, and the line is called a line of invariant points.

如果唯一解是 x = 0 且 y = 0，则原点就是唯一的不动点。如果方程退化为一条直线，那么该直线上的每个点都是不动点，这条直线就称为由不动点构成的直线。

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3. What is an invariant line? | 什么是不变线

t require each point to stay fixed; points can slide along the line. In set notation, T(L) ⊆ L, and for invertible linear transformations T(L) = L.

如果直线 L 上每个点的像点仍在该直线上，则称 L 在 M 下是不变线。这并不要求每个点保持固定；点可以沿直线滑动。用集合记号表示为 T(L) ⊆ L，对于可逆线性变换有 T(L) = L。

is a stronger condition: every point on the line is fixed, not just mapped into the line.

不变线可以经过原点，也可以是一般直线 y = mx + c 或 x = k。由不动点构成的直线是更强的条件：该直线上的每个点都保持固定，而不仅仅是被映射到这条直线上。

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4. Invariant lines through the origin | 过原点的不变线

v. Thus the direction of the line is an eigenvector and λ is the corresponding eigenvalue.

对于线性变换，过原点的不变线具有方向向量 v，使得 M v = λ v。因此，该直线的方向是矩阵 M 的特征向量，λ 是对应的特征值。

− bc = 0. For each real eigenvalue, solve (M − λI)v = 0 to find the eigenvector direction.

特征值通过求解特征方程 det(M − λI) =

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