常用矩阵导数公式

Posted 2020-10-08

1 矩阵\\(Y=f(x)\\)对标量x求导

     矩阵Y是一个\\(m\\times n\\)的矩阵，对标量x求导，相当于矩阵中每个元素对x求导

\\[\\frac{dY}{dx}=\\begin{bmatrix}\\dfrac{df_{11}(x)}{dx} & \\ldots & \\dfrac{df_{1n}(x)}{dx} \\\\ \\vdots & \\ddots &\\vdots \\\\ \\dfrac{df_{m1}(x)}{dx} & \\ldots & \\dfrac{df_{mn}(x)}{dx} \\end{bmatrix}\\]

2 标量y=f(x)对矩阵X求导

     注意与上面不同，这次括号内是求偏导，\\(X\\)是是一个\\(m\\times n\\)的矩阵，函数\\(y=f(x)\\)对矩阵\\(X\\)中的每个元素求偏导，对\\(m\\times n\\)矩阵求导后还是\\(m\\times n\\)矩阵

\\[\\frac{dy}{dX} = \\begin{bmatrix}\\dfrac{\\partial f}{\\partial x_{11}} & \\ldots & \\dfrac{\\partial f}{\\partial x_{1n}}\\\\ \\vdots & \\ddots & \\vdots \\\\\\dfrac{\\partial f}{\\partial x_{m1}} & \\ldots & \\dfrac{\\partial f}{\\partial x_{mn}}\\end{bmatrix}\\]




3 函数矩阵Y对矩阵X求导

矩阵\\(Y=F(x)\\)对每一个\\(X\\)的元素求导，构成一个超级矩阵

\\[F(x)=\\begin{bmatrix}f_{11}(x) & \\ldots &  f_{1n}(x)\\\\ \\vdots & \\ddots &\\vdots \\\\ f_{m1}(x) & \\ldots & f_{mn}(x) \\end{bmatrix}\\]

\\[X=\\begin{bmatrix}x_{11} & \\ldots &  x_{1s}\\\\ \\vdots & \\ddots &\\vdots \\\\ x_{r1} & \\ldots & x_{rs}\\end{bmatrix}\\]

，其中

\\[\\frac{dF}{dX} = \\begin{bmatrix}\\dfrac{\\partial F}{\\partial x_{11}} & \\ldots & \\dfrac{\\partial F}{\\partial x_{1s}}\\\\ \\vdots & \\ddots & \\vdots \\\\\\dfrac{\\partial F}{\\partial x_{r1}} & \\ldots & \\dfrac{\\partial F}{\\partial x_{rs}}\\end{bmatrix}\\]

其中

\\[\\frac{\\partial F}{\\partial x_{ij}} = \\begin{bmatrix}\\dfrac{\\partial f_{11}}{\\partial x_{ij}} & \\ldots & \\dfrac{\\partial f_{1n}}{\\partial x_{ij}}\\\\ \\vdots & \\ddots & \\vdots \\\\\\dfrac{\\partial f_{m1}}{\\partial x_{ij}} & \\ldots & \\dfrac{\\partial f_{mn}}{\\partial x_{ij}}\\end{bmatrix}\\]

\\[\\frac{\\partial F}{\\partial x_{ij}} = \\begin{bmatrix} \\end{bmatrix}\\]

重要结论：假设是一个向量：

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