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선형대수/ 유용한 미적분 항등식들
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선형대수/ 유용한 미적분 항등식들
Scalar to Scalar
Vector to Scalar
Matrix to Scalar
참조
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다음은 널리 사용되는 미적분의 항등식들이다.
Scalar to Scalar
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\begin{aligned} {d \over dx}cx^n &= cnx^{n-1} \\ {d \over dx} \log(x) &= 1/x \\ {d \over dx} \exp(x) &= \exp(x) \\ {d \over dx}[f(x) + g(x)] &= {df(x) \over dx} + {dg(x) \over dx} \\ {d \over dx}[f(x)g(x)] &= f(x){d g(x) \over dx} + g(x) {d f(x) \over dx} \\ {d \over dx}f(u(x)) &= {du \over dx}{df(u) \over du} \end{aligned}
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Vector to Scalar
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\begin{aligned} {\partial (\bold{a}^\top\bold{x}) \over \partial \bold{x}} &= \bold{a} \\ {\partial (\bold{b}^T\bold{Ax}) \over \partial \bold{x}} &= \bold{A}^\top\bold{b} \\ {\partial (\bold{x}^\top\bold{Ax}) \over \partial \bold{x}} &= (\bold{A} + \bold{A}^\top)\bold{x} \end{aligned}
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Matrix to Scalar
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{\partial f \over \partial \bold{X}} = \left( \begin{matrix} {\partial f \over \partial x_{11}} & ... & {\partial f \over \partial x_{1n}} \\ \vdots & \ddots & \vdots \\ {\partial f \over \partial x_{m1}} & ... & {\partial f \over \partial x_{mn}} \end{matrix} \right)
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•
이차식을 포함한 항등식
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\begin{aligned} {\partial \over \partial \bold{X}}(\bold{a}^\top{\bold{Xb}}) &= \bold{ab}^\top \\ {\partial \over \partial \bold{X}}(\bold{a}^\top{\bold{X}^\top\bold{b}}) &= \bold{ba}^\top \end{aligned}
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trace를 포함한 항등식
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\begin{aligned} {\partial \over \partial \bold{X}}\text{tr}(\bold{AXB}) &= \bold{A}^\top\bold{B}^\top \\ {\partial \over \partial \bold{X}}\text{tr}(\bold{X}^\top{\bold{A}}) &= \bold{A} \\ {\partial \over \partial \bold{X}}\text{tr}(\bold{X}^{-1}\bold{A}) &= -\bold{X}^\top\bold{A}^\top\bold{X}^{-\top} \\ {\partial \over \partial \bold{X}}\text{tr}(\bold{X}^\top\bold{AX}) &= (\bold{A+A}^\top)\bold{X} \end{aligned}
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determinant을 포함한 항등식
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\begin{aligned} {\partial \over \partial \bold{X}}\text{det}(\bold{AXB}) &= \det(\bold{AXB})\bold{B}^\top \cdot \text{adj}(\bold{AXB})\bold{A}^\top = \text{det}(\bold{AXB})\bold{X}^{-\top} \\ {\partial \over \partial \bold{X}} \log(\det(\bold{X})) &= \bold{X}^{-\top} \end{aligned}
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참조
•
Probabilistic Machine Learning: An Introduction
