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矩阵分析(九)Gram矩阵

November 9, 2020 • Read: 146 • 数学阅读设置

欧氏空间

$V$是$\mathbb{R}$上的线性空间,定义映射

$$ \sigma: V\times V \to \mathbb{R} $$

对于$\alpha, \beta \in V$,将$\sigma(\alpha, \beta)$记为$\left<\alpha, \beta\right>$,若$\sigma$满足:

  1. 对称性:$\left<\alpha.\beta\right>=\left<\beta, \alpha\right>$
  2. (右)齐次性:$\left<\alpha, k\beta\right>=k\left<\alpha,\beta\right>$
  3. (右)可加性:$\left<\alpha, \beta+\gamma\right>=\left<\alpha,\beta\right>+\left<\alpha, \gamma\right>$
  4. 非负性:$\left<\alpha,\alpha\right>≥0$,且$\left<\alpha,\alpha\right>=0\Leftrightarrow\alpha=0$

则称$\sigma$为$V$上的(实)内积,当$V$是有限维时,称其为欧氏空间($\mathbb{R}^n$为标准欧氏空间)

实际上$\alpha$是一个向量,$\beta$是一个向量,$\left<\alpha, \beta\right>$表示向量$\alpha$与向量$\beta$的内积,结果是一个实数

实内积的性质
  1. (左)齐次性:$\left<k\alpha, \beta\right>=k\left<\alpha,\beta\right>$
  2. (左)可加性:$\left<\alpha+\beta, \gamma\right>=\left<\alpha,\gamma\right>+\left<\beta, \gamma\right>$
  3. $\left<k_1\alpha_1+···+k_s\alpha_s,\beta\right>=k_1\left<\alpha_1,\beta\right>+···k_s\left<\alpha_s,\beta\right>$
  4. $\left<\alpha,k_1\beta_1+···+k_s\beta_s\right>=k_1\left<\alpha,\beta_1\right>+···k_s\left<\alpha,\beta_s\right>$

复内积

$V$是$\mathbb{C}$上的线性空间,定义映射

$$ \sigma: V\times V \to \mathbb{C} $$

对于$\alpha, \beta \in V$,将$\sigma(\alpha, \beta)$记为$\left<\alpha, \beta\right>$,若$\sigma$满足:

  1. 共轭对称性:$\left<\alpha.\beta\right>=\overline{\left<\beta, \alpha\right>}$
  2. (右)齐次性:$\left<\alpha, k\beta\right>=k\left<\alpha,\beta\right>$
  3. (右)可加性:$\left<\alpha, \beta+\gamma\right>=\left<\alpha,\beta\right>+\left<\alpha, \gamma\right>$
  4. 非负性:$\left<\alpha,\alpha\right>≥0$,且$\left<\alpha,\alpha\right>=0\Leftrightarrow\alpha=0$

则称$\sigma$为$V$上的(复)内积,当$V$是有限维时,称其为酉空间($\mathbb{R}^n$为标准欧氏空间)

复内积的性质
  1. (左)齐次性:$\left<k\alpha, \beta\right>=\bar{k}\left<\alpha,\beta\right>$
  2. (左)可加性:$\left<\alpha+\beta, \gamma\right>=\left<\alpha,\gamma\right>+\left<\beta, \gamma\right>$
  3. $\left<k_1\alpha_1+···+k_s\alpha_s,\beta\right>=\overline{k_1}\left<\alpha_1,\beta\right>+···\overline{k_s}\left<\alpha_s,\beta\right>$
  4. $\left<\alpha,k_1\beta_1+···+k_s\beta_s\right>=k_1\left<\alpha,\beta_1\right>+···k_s\left<\alpha,\beta_s\right>$

线性组合的内积的矩阵表示

$\alpha_1,...,\alpha_s;\beta_1,...,\beta_t$是$\mathbb{C}$上的内积空间$V$中的两个向量组,则

$$ \begin{aligned} \left<k_1\alpha_1+···+k_s\alpha_s,l_1\beta_1+···+l_t\beta_t\right>\\ =(\overline{k_1},...,\overline{k_s})\begin{bmatrix}\left<\alpha_1,\beta_1\right>&\cdots &\left<\alpha_1,\beta_t\right>\\ \vdots & \ddots &\vdots \\\left<\alpha_s,\beta_1\right> &\cdots & \left<\alpha_s,\beta_t\right>\end{bmatrix}\begin{bmatrix}l_1\\ \vdots \\ l_t\end{bmatrix} \end{aligned} $$


Gram矩阵

$\alpha_1,...,\alpha_s;\beta_1,...,\beta_t$是$\mathbb{C}$上的内积空间$V$中的两个向量组,则

$$ \begin{bmatrix}\left<\alpha_1,\beta_1\right>&\cdots &\left<\alpha_1,\beta_t\right>\\ \vdots & \ddots &\vdots \\\left<\alpha_s,\beta_1\right> &\cdots & \left<\alpha_s,\beta_t\right>\end{bmatrix} $$

称为$\alpha_1,...,\alpha_s;\beta_1,...,\beta_t$的协Gram矩阵,记为$G(\alpha_1,...,\alpha_s;\beta_1,...,\beta_t)$

$\alpha_1,...,\alpha_s$是$\mathbb{C}$上的内积空间$V$中的一个向量组,则

$$ \begin{bmatrix}\left<\alpha_1,\beta_1\right>&\cdots &\left<\alpha_1,\beta_t\right>\\ \vdots & \ddots &\vdots \\\left<\alpha_s,\beta_1\right> &\cdots & \left<\alpha_s,\beta_t\right>\end{bmatrix} $$

称为$\alpha_1,...,\alpha_s$的Gram矩阵,记为$G(\alpha_1,...,\alpha_s)$

$\alpha_1,...,\alpha_s$是$\mathbb{C}^n$中的一个向量组,记$A=(\alpha_1,...,\alpha_s)$,则

$$ G(\alpha_1,...,\alpha_s)=A^HA $$

其中,$A^H=(\bar{A})^T=\overline{(A^T)}$

$\alpha_1,...,\alpha_s$是$\mathbb{R}^n$中的一个向量组,记$A=(\alpha_1,...,\alpha_s)$,则

$$ G(\alpha_1,...,\alpha_s)=A^TA $$

$\alpha_1,...,\alpha_s;\beta_1,...,\beta_t$是$\mathbb{C}$上的内积空间$V$中的两个向量组,如果$\alpha_1,...,\alpha_s$可由$\beta_1,...,\beta_t$线性表出,且

$$ (\alpha_1,...,\alpha_s)=(\beta_1,...,\beta_t)A $$

$$ G(\alpha_1,...,\alpha_s)=A^HG(\beta_1,...,\beta_t)A $$

Gram矩阵的性质
  1. $Rank(G)=rank(\alpha_1,...,\alpha_s)$
  2. Hermite性:$G^H=G$
  3. 非负性:$\forall x\in \mathbb{C}^s$,复二次型$x^HGx≥0$,并且$G$正定$\Leftrightarrow \alpha_1,...,\alpha_s$线性无关
Last Modified: November 25, 2020
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