📚 Eigenvectors and Eigenvalues | 特征向量与特征值
Eigenvectors and eigenvalues appear when a square matrix acts as a linear transformation. Instead of describing how every point moves, we ask a simpler question: are there any non-zero vectors whose direction is left unchanged by the transformation? These special directions are called eigenvectors, and the scale factors attached to them are eigenvalues.
当方阵作为线性变换时，特征向量和特征值就会出现。我们不再描述每个点如何移动，而是问一个更简单的问题：是否存在某些非零向量，其方向在变换下保持不变？这些特殊的方向称为特征向量，与之对应的缩放因子称为特征值。
1. What Are Eigenvectors and Eigenvalues? | 什么是特征向量与特征值
mn vector v is an eigenvector of A if A v is a scalar multiple of v. The scalar λ is the corresponding eigenvalue.
设 A 为 n × n 矩阵。如果非零列向量 v 满足 A v 是 v 的标量倍，则 v 是 A 的特征向量。该标量 λ 就是对应的特征值。
A v = λ v
This means the transformation A does not rotate v; it only stretches, shrinks, or reverses it.
这意味着变换 A 不会旋转 v；它只会拉伸、缩短或反向伸缩 v。
ctors must be non-zero, because A 0 = λ 0 is true for every λ and tells us nothing useful.
特征向量必须是非零的，因为 A 0 = λ 0 对任意 λ 都成立，不能提供有用信息。
2. The Defining Equation A v = λ v | 定义方程 A v = λ v
v − λ v = 0. Since λ v = λ I v, where I is the identity matrix, we obtain (A − λ I) v = 0.
将 A v = λ v 变形为 A v − λ v = 0。因为 λ v = λ I v，其中 I 是单位矩阵，我们得到 (A − λ I) v = 0。
t, the matrix A − λ I must be singular, so its determinant must be zero: det(A − λ I) = 0.
要使非零向量 v 存在，矩阵 A − λ I 必须是奇异的，因此其行列式必须为零：det(A − λ I) = 0。
This equation is called the characteristic equation. Its left-hand side is the characteristic polynomial in λ.
这个方程称为特征方程。它的左边是关于 λ 的特征多项式。
| Key form | A v = λ v |
| Rearranged | (A − λ I) v = 0 |
| Condition | det(A − λ I) = 0 |
The table summarises the three equivalent starting points used in AQA exam questions.
该表总结了 AQA 考题中常用的三个等价出发点。
3. Finding Eigenvalues Using the Characteristic Equation | 用特征方程求特征值
matrix A = [[a, b], [c, d]], the characteristic equation is λ² − (a + d)λ + (ad − bc) = 0.
对于 2 × 2 矩阵 A = [[a, b], [c, d]]，特征方程为 λ² − (a + d)λ + (ad − bc) = 0。
λ² − (a + d)λ + (ad − bc) = 0
The coefficient a + d is the trace of A, and ad − bc is the determinant of A. This often gives a quick check: sum of eigenvalues equals trace, product equals determinant.
系数 a + d 是 A 的迹，ad − bc 是 A 的行列式。这提供了一个快速检验：特征值之和等于迹，特征值之积等于行列式。
Write det(A − λ I) carefully. A common sign error is to compute det(λ I − A) or det(A + λ I) without adjusting signs.
请仔细写出 det(A − λ I)。常见符号错误是计算 det(λ I − A) 或 det(A + λ I) 时没有调整符号。
4. Worked Example: Eigenvalues of a 2 × 2 Matrix | 例题：2 × 2 矩阵的特征值
Find the eigenvalues of A = [[2, 1], [1, 2]].
求矩阵 A = [[2, 1], [1, 2]] 的特征值。
A − λ I = [[2 − λ, 1], [1, 2 − λ]]. Then det(A − λ I) = (2 − λ)(2 − λ) − 1 = (2 − λ)² − 1.
构造 A − λ I = [[2 − λ, 1], [1, 2 − λ]]。则 det(A − λ I) = (2 − λ)(2 − λ) − 1 = (2 − λ)² − 1。
Expand: (4 − 4λ + λ²) − 1 = λ² − 4λ + 3. Set equal to zero: λ² − 4λ + 3 = 0.
展开：(4 − 4λ + λ²) − 1 = λ² − 4λ + 3。令其为零：λ² − 4λ + 3 = 0。
Factorise: (λ − 1)(λ − 3) = 0, so λ₁ = 1 and λ
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