矩阵分析与应用+张贤达_标量函数的梯度矩阵 公式

1. 标量函数的共轭梯度公式

(1)若$f(x)=c$为常数，则共轭梯度$\frac{\partial c}{\partial x^{\ast }}=0$。
(2)线性法则:若$f(x)$和$g(x)$分别是向量$x$的实值函数，$c_1$和$c_2$为复常数，则
$$\frac{\partial [c_{1}f(x)+c_{2}g(x)]}{\partial x^{\ast }}=c_1\frac{\partial f(x)}{\partial x^{\ast }}+c_2\frac{\partial g(x)}{\partial x^{\ast }}$$
(3)乘积法则:

• 若$f(x)$和$g(x)$都是向量$x$的实值函数，则
  $$\frac{\partial f(x)g(x)}{\partial x^{\ast }}=g(x)\frac{\partial f(x)}{\partial x^{\ast }}+f(x)\frac{\partial g(x)}{\partial x^{\ast }}$$
• 若$f(x),g(x)$和$h(x)$都是向量$x$的实值函数，则
  $$\frac{\partial f(x)g(x)h(x)}{\partial x^{\ast }}=g(x)h(x)\frac{\partial f(x)}{\partial x^{\ast }}+f(x)h(x)\frac{\partial g(x)}{\partial x^{\ast }}+f(x)g(x)\frac{\partial h(x)}{\partial x^{\ast }}$$
  (4)商法则:若$g(x)\ne 0$，则
  $$\frac{\partial f(x)/g(x)}{\partial x^{\ast }}=\frac{1}{g^2(x)}[g(x)\frac{\partial f(x)}{\partial x^{\ast }}-f(x)\frac{\partial g(x)}{\partial x^{\ast }}]$$
  (5)链式法则：若$y(x)$是$x$的复向量值函数，则
  $$\frac{\partial f(y(x))}{\partial x^{\ast }}=\frac{\partial [y(x){]}^T}{\partial x^{\ast }}\frac{\partial f(y)}{\partial y}$$
  式中，$\frac{\partial [y(x){]}^T}{\partial x^{\ast }}$为$n\times n$矩阵。
  (6)若$n\times 1$向量为$a$与$x$无关的常数向量，则
  $$\frac{\partial a^{H}x}{\partial x^{\ast }}=0,\frac{\partial x^{H}a}{\partial x^{\ast }}=a$$
  (7)令$A$是一个与向量$x$无关的矩阵，则
  $$\frac{\partial x^{H}Ax}{\partial x}=A^T{x}^{\ast },\frac{\partial x^{H}Ax}{\partial x^{\ast }}=Ax$$
  $$\frac{\partial x^{H}Ay}{\partial A}=x^{\ast }{y}^T,\frac{\partial x^{H}Ax}{\partial A}=x^{\ast }{x}^T$$

2. 奇异值分解

矩阵$A$的奇异值应该能够描述$A$的奇异性质。
定理
令$A\in C^{m\times n}(m>n)$的奇异值为
$${\sigma }_1\ge {\sigma }_2\ge \cdot \cdot \cdot \ge {\sigma }_n\ge 0$$
则
$${\sigma }_k=\underset{E\in C^{m\times n}}{\min }||E|{|}_F:rank(A+E)\ne k-1,k=1,2,\cdot \cdot \cdot ,n$$
并且存在一满足$||E_k|{|}_F={\sigma }_k$的误差矩阵$E$使得
$$rank(A+E_k)=k-1,k=1,2,\cdot \cdot \cdot ,n$$
定义表明，奇异值与使得原矩阵$A$的秩减小1的误差矩阵$E_k$的Frobenius范数相等。