# 高等代数选讲笔记（13）——张量

Contents

## 第十三讲：张量

### 多线性映射

#### 定义

$V_1, \cdots, V_p, V$ 为向量空间，一个的多线性映射为：
$$F : V_1 \times V_2 \times \cdots \times V_p \to V$$

$$F(v_1 + cv_1′, v_2, \cdots, v_p) = F(v_1, \cdots, v_p) + cF(v_1′, v_2, \cdots, v_p)$$

#### 多线性映射集合

$$\mathcal{L}(V_1, \cdots, V_p; V) \not = \mathcal{L}(V_1 \times \cdots \times V_P; V)$$

$$f_1 \otimes \cdots \otimes f_p(v_1, \cdots, v_p) = f_1(v_1) \cdot \cdots \cdot f_p(v_p), \forall v_1 \in V_1, \cdots, v_p \in V_p \\ V_1 \times \cdots \times V_p \to \mathbb{C}$$

$$f_1 \otimes \cdots \otimes f_p \in \mathcal{L}(V_1, \cdots, V_p; \mathbb{C})$$

#### 性质 1

$$\{\tilde{v}_i \otimes \tilde{w}_j\}_{i, j}$$

$\forall F \in L(V, W; \mathbb{C})$，$(v_1, \cdots, v_m) \subseteq V$ 为 $(\tilde{v}_1, \cdots, \tilde{v}_m)$ 的对偶基，$(w_1, \cdots, w_n)$ 为 $(\tilde{w}_1, \cdots, \tilde{w}_n)$ 的对偶基。

$$G = \sum_{i, j} \alpha_{ij} \tilde{v}_i \otimes \tilde{w}_j \in \mathcal{L}(V, W; \mathbb{C}) \\ F((v_i, w_j)) = \alpha_{ij} = G((v_i, w_j)) \Rightarrow F = G$$

$$\sum_{i, j} C_{ij} \tilde{v}_i \otimes \tilde{w}_j = 0$$

$$\sum_{i, j} C_{ij} \tilde{v}_i \otimes \tilde{w}_j((v_k, w_l)) \\ = C_{k, l} = 0 \\ \Rightarrow C_{ij} = 0, \forall i, j$$

#### 性质 2

$$B_i = \{b_1^i, b_2^i, \cdots, b_{\dim V_i}^i\}$$

$$\{b_{j_1}^{1*} \otimes \cdots \otimes b_{j_p}^{p*}\}_{j_1, \cdots, j_p}$$

$$\dim \mathcal{L}(V_1, \cdots, V_p; \mathbb{C}) = \dim V_1 \cdot \cdots \cdot \dim V_p$$

### 张量积

#### 定义

$V_1, \cdots, V_P$ 为向量空间，它们的张量积为：
$$V_1 \otimes V_2 \cdots \otimes V_p := \mathcal{L}(V_1^*, V_2^*, \cdots, V_p^*; \mathbb{C})$$

$v_1 \in V_1, \cdots, v_p \in V_p$，那么：
$$v_1 \otimes \cdots \otimes v_p \in V_1 \otimes \cdots \otimes V_p$$

$$\dim (V_1 \otimes \cdots \otimes V_p) = \dim V_1 \cdot \cdots \cdot \dim V_p$$
：任意 $t \in V_1 \otimes \cdots \otimes V_p$ 可以写成：
$$c_1 v_1^1 \otimes \cdots \otimes v_p^1 + \cdots + c_r v_1^r \otimes \cdots \otimes v_p^r$$

$$v_1 \otimes v_2 \otimes \cdots \otimes v_p$$

#### 张量的泛性质

$$T : V_1 \otimes \cdots \otimes V_p \to V$$

$$F(v_1, \cdots, v_p) = T(v_1 \otimes \cdots \otimes v_p), \forall v_k \in V_k, k = 1, \cdots, p$$

$$\{b_{j_1}^1 \otimes b_{j_2}^2 \otimes \cdots \otimes b_{j_p}^p \}_{j_1, \cdots, j_p}$$

$$T : V_1 \otimes \cdots \otimes V_p \to V \\ b_{j_1}^1 \otimes \cdots \otimes b_{j_p}^p \mapsto F(b_{j_1}^1, \cdots, b_{j_p}^p)$$
$T$ 满足对于任意 $v_1, \cdots, v_p$：
$$v_k = \sum_{j_k} \alpha_{j_k}^k b_{j_k}^k, k = 1, \cdots, p$$

$$T(v_1 \otimes \cdots \otimes v_p) = \sum_{j_1, \cdots, j_p} \alpha_{j_1}^1 \cdots \alpha_{j_p}^p T(b_{j_1}^1 \otimes \cdots \otimes b_{j_p}^p)$$

$$F(v_1, \cdots, v_p) = \sum_{j_1, \cdots, j_p} \alpha_{j_1}^1 \cdots \alpha_{j_p}^p F(b_{j_1}^1, \cdots, b_{j_p}^p)$$

$$\mathcal{L}(V_1, \cdots, V_p; V) = \mathcal{L}(V_1 \otimes \cdots \otimes V_p; V) \\ F \mapsto T$$

#### 性质

$$T :V^* \otimes W \to \mathcal{L}(V; W)$$

$$F : V^* \times W \to \mathcal{L}(V; W) \\ (l, w) \mapsto \{v \mapsto l(v) \cdot w\}$$

$\mathcal{L}(V, W)$ 可以由 $\{v \mapsto l(v) \cdot w\}_{v \in V, w \in W}$ 生成 $\Rightarrow T$ 是满射。

$$V^* \times V \to C \\ (l, v) \mapsto l(v)$$

$$\varepsilon : V^* \otimes V \to \mathbb{C}$$

$$\eta : \mathbb{C} \to \mathcal{L}(V; V) \\ c \mapsto c I_V$$

$$\mathbb{C} \stackrel{\eta}{\to} \mathcal{L}(V; V) \stackrel{T^{-1}}{\to} V^* \otimes V \stackrel{\varepsilon}{\to} \mathbb{C}$$

### 张量积的矩阵

#### 性质

$$T_1 : V_1 \to W_1, T_2 : V_2 \to W_2$$

$$T_1 \otimes T_2 : V_1 \otimes V_2 \to W_1 \otimes W_2 \\ v_1 \otimes v_2 \mapsto T_1(v_1) \otimes T_2(v_2)$$

$$[T_1]_{A_1, B_1} \otimes [T_2]_{A_2, B_2} = [T_1 \otimes T_2]_{A_1 \otimes A_2, B_1 \otimes B_2}$$

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, B = \begin{pmatrix} e & f \\ g & h \end{pmatrix} \\ A \otimes B = \begin{pmatrix} aB & bB \\ cB & dB \end{pmatrix} = \begin{pmatrix} ae & af & be & bf \\ ag & ah & bg & bh \\ ce & cf & de & df \\ cg & ch & dg & dh \end{pmatrix}$$

$$(A_1 \otimes A_2)_{(i_1, i_2), (j_1, j_2)} = (A_1)_{i_1, j_1} (A_2)_{i_2, j_2}$$