特征值与特征向量

特征值和特征向量揭示了矩阵最本质的性质——它在哪些方向上只做拉伸而不改变方向（Av = λv）。在 AI 中：PCA 降维对协方差矩阵做特征值分解，取最大特征值对应的主成分，广泛用于图像压缩、人脸识别（Eigenface）、高维数据可视化（t-SNE 预处理常先 PCA 到 50 维）；谱聚类对图拉普拉斯矩阵求最小特征向量实现非凸聚类；Google PageRank 求转移矩阵的主特征向量来排序网页；协方差矩阵的特征值分布判断特征共线性——条件数（最大/最小特征值之比）过高意味着梯度不稳定；奇异值分解（SVD）与特征值分解紧密关联，用于推荐系统矩阵分解、NLP 词向量训练（GloVe）。本节从 2×2 特征多项式出发，到完整 PCA 流程，每一步用具体数字说明。

一、定义：什么是特征值与特征向量

二、手动计算特征值与特征向量（完整推导）

方法：求解 det(A - λI) = 0（特征方程），然后代入求特征向量。

三、特征值分解（对角化）

四、对称矩阵的特殊性质

五、AI 核心应用——PCA 主成分分析

PCA 是最经典的降维方法，核心就是对协方差矩阵做特征值分解。

六、其他 AI 应用

七、代码验证（C# / Rust）

C#（.NET 10）

// dotnet run 即可执行
// 2×2 特征值分解: 特征多项式 λ²-trace·λ+det=0
void Eig2(double a, double b, double c, double d,
          out double l1, out double l2, out double[] v1, out double[] v2)
{
    double tr = a + d, det = a * d - b * c;
    double disc = Math.Sqrt(tr * tr - 4 * det);
    l1 = (tr + disc) / 2; l2 = (tr - disc) / 2;
    // 特征向量: (A-λI)v=0, v ∝ [b, λ-a]
    if (Math.Abs(b) > 1e-12) {
        v1 = new[] { b, l1 - a }; v2 = new[] { b, l2 - a };
    } else { v1 = new[] { 1.0, 0.0 }; v2 = new[] { 0.0, 1.0 }; }
    double n1 = Math.Sqrt(v1[0]*v1[0]+v1[1]*v1[1]);
    double n2 = Math.Sqrt(v2[0]*v2[0]+v2[1]*v2[1]);
    v1[0] /= n1; v1[1] /= n1; v2[0] /= n2; v2[1] /= n2;
}

// ===== 基本特征值分解 =====
Eig2(4, 1, 2, 3, out double lam1, out double lam2, out double[] ev1, out double[] ev2);
Console.WriteLine($"特征值: [{lam1}, {lam2}]");
Console.WriteLine($"特征向量: [{ev1[0]:F4},{ev1[1]:F4}], [{ev2[0]:F4},{ev2[1]:F4}]");
Console.WriteLine($"Av1 = [{4*ev1[0]+1*ev1[1]:F4}, {2*ev1[0]+3*ev1[1]:F4}]");
Console.WriteLine($"λ1·v1= [{lam1*ev1[0]:F4}, {lam1*ev1[1]:F4}]");
Console.WriteLine($"Av2 = [{4*ev2[0]+1*ev2[1]:F4}, {2*ev2[0]+3*ev2[1]:F4}]");
Console.WriteLine($"λ2·v2= [{lam2*ev2[0]:F4}, {lam2*ev2[1]:F4}]");

// ===== 完整 PCA 流程 =====
double[,] data = {{90,85},{70,60},{80,75},{60,55},{50,45}};
int n = 5;
double[] mu = new double[2];
for (int j = 0; j < 2; j++) { for (int i = 0; i < n; i++) mu[j] += data[i,j]; mu[j] /= n; }

double[,] cen = new double[5,2];
for (int i = 0; i < n; i++) for (int j = 0; j < 2; j++) cen[i,j] = data[i,j] - mu[j];

// 样本协方差 (ddof=1)
double[,] cov = new double[2,2];
for (int j1 = 0; j1 < 2; j1++) for (int j2 = 0; j2 < 2; j2++) {
    for (int i = 0; i < n; i++) cov[j1,j2] += cen[i,j1]*cen[i,j2];
    cov[j1,j2] /= n - 1;
}
Console.WriteLine($"\n协方差矩阵:\n  [{cov[0,0]:F2}, {cov[0,1]:F2}]\n  [{cov[1,0]:F2}, {cov[1,1]:F2}]");

Eig2(cov[0,0],cov[0,1],cov[1,0],cov[1,1], out double e1, out double e2, out double[] pc1, out _);
Console.WriteLine($"特征值: [{e1:F2}, {e2:F2}]");
Console.WriteLine($"方差解释比: [{e1/(e1+e2):F4}, {e2/(e1+e2):F4}]");

Console.Write("第1主成分得分: ");
for (int i = 0; i < n; i++) Console.Write($"{cen[i,0]*pc1[0]+cen[i,1]*pc1[1]:F2} ");
Console.WriteLine();

// ===== 对称矩阵验证正交性 =====
Eig2(5, 2, 2, 2, out _, out _, out double[] u1, out double[] u2);
Console.WriteLine($"\n对称矩阵 [[5,2],[2,2]] 特征向量内积: {u1[0]*u2[0]+u1[1]*u2[1]:F10}");

Rust

// 2×2 特征值分解
fn eig2(a: f64, b: f64, c: f64, d: f64) -> (f64, f64, [f64;2], [f64;2]) {
    let (tr, det) = (a + d, a.mul_add(d, -(b * c)));
    let disc = tr.mul_add(tr, -4.0*det).sqrt();
    let (l1, l2) = ((tr+disc)/2.0, (tr-disc)/2.0);
    let (mut v1, mut v2) = if b.abs() > 1e-12 {
        ([b, l1-a], [b, l2-a])
    } else { ([1.0, 0.0], [0.0, 1.0]) };
    let n1 = v1[0].mul_add(v1[0], v1[1]*v1[1]).sqrt();
    let n2 = v2[0].mul_add(v2[0], v2[1]*v2[1]).sqrt();
    v1[0]/=n1; v1[1]/=n1; v2[0]/=n2; v2[1]/=n2;
    (l1, l2, v1, v2)
}

fn main() {
    // ===== 基本特征值分解 =====
    let (l1, l2, v1, v2) = eig2(4.0, 1.0, 2.0, 3.0);
    println!("特征值: [{l1}, {l2}]");
    println!("特征向量: [{:.4},{:.4}], [{:.4},{:.4}]", v1[0],v1[1], v2[0],v2[1]);
    println!("Av1 = [{:.4}, {:.4}]", 4.0_f64.mul_add(v1[0], v1[1]), 2.0_f64.mul_add(v1[0], 3.0*v1[1]));
    println!("λ1·v1= [{:.4}, {:.4}]", l1*v1[0], l1*v1[1]);
    println!("Av2 = [{:.4}, {:.4}]", 4.0_f64.mul_add(v2[0], v2[1]), 2.0_f64.mul_add(v2[0], 3.0*v2[1]));
    println!("λ2·v2= [{:.4}, {:.4}]", l2*v2[0], l2*v2[1]);

    // ===== PCA =====
    let data = [[90.0,85.0],[70.0,60.0],[80.0,75.0],[60.0,55.0],[50.0,45.0_f64]];
    let n = 5;
    let mut mu = [0.0_f64; 2];
    for j in 0..2 { for i in 0..n { mu[j] += data[i][j]; } mu[j] /= n as f64; }

    let mut cen = [[0.0_f64;2];5];
    for i in 0..n { for j in 0..2 { cen[i][j] = data[i][j]-mu[j]; } }

    let mut cov = [[0.0_f64;2];2];
    for j1 in 0..2 { for j2 in 0..2 {
        for i in 0..n { cov[j1][j2] += cen[i][j1]*cen[i][j2]; }
        cov[j1][j2] /= (n-1) as f64;
    }}
    println!("\n协方差矩阵:\n  [{:.2}, {:.2}]\n  [{:.2}, {:.2}]",
             cov[0][0],cov[0][1],cov[1][0],cov[1][1]);

    let (e1, e2, pc1, _) = eig2(cov[0][0],cov[0][1],cov[1][0],cov[1][1]);
    println!("特征值: [{e1:.2}, {e2:.2}]");
    println!("方差解释比: [{:.4}, {:.4}]", e1/(e1+e2), e2/(e1+e2));

    print!("第1主成分得分: ");
    for i in 0..n { print!("{:.2} ", cen[i][0].mul_add(pc1[0], cen[i][1]*pc1[1])); }
    println!();

    // ===== 对称矩阵正交性 =====
    let (_, _, u1, u2) = eig2(5.0, 2.0, 2.0, 2.0);
    println!("\n对称矩阵 [[5,2],[2,2]] 特征向量内积: {:.10}",
             u1[0].mul_add(u2[0], u1[1]*u2[1]));
}