## 特征值

### 特征值的分解

Aυ = λυ。

AV = VΛ。

A = VΛV–1

```A = 0 -6 -1 6 2 -16 -5 20 -10```

`lambda = eig(A)`

```lambda = -3.0710 -2.4645+17.6008i -2.4645-17.6008i```

```[V,D] = eig(A) ```
```V = -0.8326 0.2003 - 0.1394i 0.2003 + 0.1394i -0.3553 -0.2110 - 0.6447i -0.2110 + 0.6447i -0.4248 -0.6930 -0.6930 D = -3.0710 0 0 0 -2.4645+17.6008i 0 0 0 -2.4645-17.6008i```

### 多重特征值

```A = [ 1 -2 1 0 1 4 0 0 3 ]```

`[V,D] = eig(A)`

```V = 1.0000 1.0000 -0.5571 0 0.0000 0.7428 0 0 0.3714 D = 1 0 0 0 1 0 0 0 3```

λ =1 时有一个双精度特征值。`V` 的第一列和第二列相同。对于此矩阵，并不存在一组完整的线性无关特征向量。

### 舒尔分解

A = USU ′ ,

```A = [ 6 12 19 -9 -20 -33 4 9 15 ]; [V,D] = eig(A)```
```V = -0.4741 + 0.0000i -0.4082 - 0.0000i -0.4082 + 0.0000i 0.8127 + 0.0000i 0.8165 + 0.0000i 0.8165 + 0.0000i -0.3386 + 0.0000i -0.4082 + 0.0000i -0.4082 - 0.0000i D = -1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 1.0000 - 0.0000i```
`[U,S] = schur(A)`
```U = -0.4741 0.6648 0.5774 0.8127 0.0782 0.5774 -0.3386 -0.7430 0.5774 S = -1.0000 20.7846 -44.6948 0 1.0000 -0.6096 0 0.0000 1.0000```

```eig(S(2:3,2:3)) ```
```ans = 1.0000 + 0.0000i 1.0000 - 0.0000i```