Chapter 7: machine learning

7.1 Shai Ben-David

https://www.youtube.com/playlist?list=PLPW2keNyw-usgvmR7FTQ3ZRjfLs5jT4BO

7.2 deep learning

7.2.1 我妻幸長

Esc = Einstein summation convention

$$\begin{aligned} W\boldsymbol{x}= & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}\sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}\overset{\text{Esc}}{=}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\\ \boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x}\\ \boldsymbol{y}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}= & \left(w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\right)^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\left[\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right]^{\intercal}=\left[W\boldsymbol{x}\right]^{\intercal}\\ = & x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\boldsymbol{x}^{\intercal}W^{\intercal}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal} \end{aligned}$$

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$$\begin{aligned} W\boldsymbol{x}= & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}\sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}\overset{\text{Esc}}{=}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =W\boldsymbol{x}= & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}\overset{\text{Esc}}{=}\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=b_{{\scriptscriptstyle \mu}}+w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}\\ \boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{y}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}= & \left(w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\right)^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\left[\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right]^{\intercal}=\left[W\boldsymbol{x}\right]^{\intercal}\\ = & x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\boldsymbol{x}^{\intercal}W^{\intercal},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ = & b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle \mu j}}^{\intercal}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\boldsymbol{x}^{\intercal}W^{\intercal}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}=b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle \mu j}}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal} \end{aligned}$$

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$$\begin{aligned} W\boldsymbol{x}= & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}\sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ \sum\limits _{\nu=0}^{n}w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}\overset{\text{Esc}}{=}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =W\boldsymbol{x}= & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+\sum\limits _{j=1}^{n}w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}\overset{\text{Esc}}{=}\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=b_{{\scriptscriptstyle \mu}}+w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}\\ \boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{y}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=y_{{\scriptscriptstyle \mu}}^{\intercal}= & \left(w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\right)^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\left[\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right]^{\intercal}=\left[W\boldsymbol{x}\right]^{\intercal}\\ = & x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\boldsymbol{x}^{\intercal}W^{\intercal}\\ = & b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle \mu j}}^{\intercal}=b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}=b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal} \end{aligned}$$

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$$\begin{aligned} \boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\boldsymbol{y}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=y_{{\scriptscriptstyle \mu}}= & w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal}\\ =\boldsymbol{x}^{\intercal}W^{\intercal}=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}=b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}=\boldsymbol{y}^{\intercal} \end{aligned}$$

---

$$\begin{aligned} \sigma\left(\boldsymbol{y}\right)=\sigma\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma\left(y_{{\scriptscriptstyle \mu}}\right)= & \sigma\left(w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\right)=\sigma\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\sigma\left(\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right)=\sigma\left(W\boldsymbol{x}\right),\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\sigma\left(\boldsymbol{y}\right)=\sigma\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma\left(y_{{\scriptscriptstyle \mu}}\right)= & \sigma\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right)=\sigma\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\sigma\left(\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right)=\sigma\left(W\boldsymbol{x}\right),\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \left(\boldsymbol{x}^{\intercal}W^{\intercal}\right)\varsigma=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}\varsigma=\left(x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\right)\varsigma=\left(x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\right)\varsigma=\left(y_{{\scriptscriptstyle \mu}}^{\intercal}\right)\varsigma=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma=\left(\boldsymbol{y}^{\intercal}\right)\varsigma\\ =\left(\boldsymbol{x}^{\intercal}W^{\intercal}\right)\varsigma=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}\varsigma=\left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma=\left(y_{{\scriptscriptstyle \mu}}^{\intercal}\right)\varsigma=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma=\left(\boldsymbol{y}^{\intercal}\right)\varsigma \end{aligned}$$

---

$$\begin{aligned} \sigma\left(\boldsymbol{y}\right)=\sigma\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma\left(y_{{\scriptscriptstyle \mu}}\right)= & \sigma\left(w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}\right)=\sigma\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\sigma\left(\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right)=\sigma\left(W\boldsymbol{x}\right),\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\sigma\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\sigma\left(\boldsymbol{y}\right)=\sigma\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma\left(y_{{\scriptscriptstyle \mu}}\right)= & \sigma\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right)=\sigma\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\sigma\left(\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}\right)=\sigma\left(W\boldsymbol{x}\right),\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ =\sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right)=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\sigma\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \left(\boldsymbol{x}^{\intercal}W^{\intercal}\right)\varsigma=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}\varsigma=\left(x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\right)\varsigma=\left(x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\right)\varsigma=\left(y_{{\scriptscriptstyle \mu}}^{\intercal}\right)\varsigma=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma=\left(\boldsymbol{y}^{\intercal}\right)\varsigma\\ =\boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}\varsigma=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma\\ =\left(\boldsymbol{x}^{\intercal}W^{\intercal}\right)\varsigma=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}\varsigma=\left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma=\left(y_{{\scriptscriptstyle \mu}}^{\intercal}\right)\varsigma=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma=\left(\boldsymbol{y}^{\intercal}\right)\varsigma\\ =\boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}\varsigma=\left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma \end{aligned}$$

---

$$\begin{aligned} \sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\sigma\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right)=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\sigma\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}\varsigma=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma\\ =\boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}\varsigma=\left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma \end{aligned}$$

---

$$\begin{aligned} \boldsymbol{z}=\sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}=\sigma\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =\boldsymbol{z}=z_{{\scriptscriptstyle \mu}}=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle \mu}} \end{pmatrix}=\sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right)=\sigma_{{\scriptscriptstyle \mu}}\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}=\sigma\begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\sigma W\boldsymbol{x},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}w_{{\scriptscriptstyle 0\nu}}x_{{\scriptscriptstyle \nu}}\\ w_{{\scriptscriptstyle 1\nu}}x_{{\scriptscriptstyle \nu}}\\ \vdots\\ w_{{\scriptscriptstyle m\nu}}x_{{\scriptscriptstyle \nu}} \end{pmatrix}^{\intercal}\varsigma=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma=\boldsymbol{z}^{\intercal}\\ =\boldsymbol{x}^{\intercal}W^{\intercal}\varsigma=\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal} & \begin{pmatrix}b_{{\scriptscriptstyle 0}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ b_{{\scriptscriptstyle 1}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}^{\intercal}\varsigma=\begin{pmatrix}b_{{\scriptscriptstyle 0}}+w_{{\scriptscriptstyle 0j}}x_{{\scriptscriptstyle j}}\\ b_{{\scriptscriptstyle 1}}+w_{{\scriptscriptstyle 1j}}x_{{\scriptscriptstyle j}}\\ \vdots\\ b_{{\scriptscriptstyle m}}+w_{{\scriptscriptstyle mj}}x_{{\scriptscriptstyle j}} \end{pmatrix}^{\intercal}\varsigma=\left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=\begin{pmatrix}y_{{\scriptscriptstyle 0}}\\ y_{{\scriptscriptstyle 1}}\\ \vdots\\ y_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=z_{{\scriptscriptstyle \mu}}^{\intercal}=\boldsymbol{z}^{\intercal} \end{aligned}$$

---

$$\begin{aligned} \boldsymbol{z}= & \sigma\boldsymbol{y}=\sigma_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}x_{{\scriptscriptstyle \nu}}=\sigma W\boldsymbol{x},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle \mu0}}=b_{{\scriptscriptstyle \mu}} \end{cases}\\ =z_{{\scriptscriptstyle \mu}}= & \sigma_{{\scriptscriptstyle \mu}}y_{{\scriptscriptstyle \mu}}=\sigma_{{\scriptscriptstyle \mu}}\left(w_{{\scriptscriptstyle \mu j}}x_{{\scriptscriptstyle j}}+b_{{\scriptscriptstyle \mu}}\right),\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \mu}}=w_{{\scriptscriptstyle \mu0}} \end{cases}\\ \boldsymbol{x}^{\intercal}W^{\intercal}\varsigma= & x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=x_{{\scriptscriptstyle \nu}}^{\intercal}w_{{\scriptscriptstyle \nu\mu}}\varsigma_{{\scriptscriptstyle \mu}}=\boldsymbol{y}^{\intercal}\varsigma=\boldsymbol{z}^{\intercal}\\ = & \left(b_{{\scriptscriptstyle \mu}}^{\intercal}+x_{{\scriptscriptstyle j}}^{\intercal}w_{{\scriptscriptstyle j\mu}}\right)\varsigma_{{\scriptscriptstyle \mu}}=y_{{\scriptscriptstyle \mu}}^{\intercal}\varsigma_{{\scriptscriptstyle \mu}}=z_{{\scriptscriptstyle \mu}}^{\intercal} \end{aligned}$$

---

matrix calculus^[57]

4-15

wrong or incompatible transpose

$$\begin{aligned} \boldsymbol{x}^{\intercal}W= & \begin{pmatrix}x_{{\scriptscriptstyle 0}} & x_{{\scriptscriptstyle 1}} & \cdots & x_{{\scriptscriptstyle m}}\end{pmatrix}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\\ = & \begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}=\begin{pmatrix}\sum\limits _{\mu=1}^{m}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu0}}\\ \sum\limits _{\mu=1}^{m}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu1}}\\ \vdots\\ \sum\limits _{\mu=1}^{m}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu n}} \end{pmatrix}^{\intercal}\\ \overset{\text{Einstein summation convention}}{=} & \begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}=\begin{pmatrix}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu0}}\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu1}}\\ \vdots\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu n}} \end{pmatrix}^{\intercal}\\ = & x_{{\scriptscriptstyle \mu}}^{\intercal}w_{{\scriptscriptstyle \mu\nu}}=\left(x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}\right)^{\intercal}? \end{aligned}$$

4-18

wrong or incompatible transpose

$$\begin{aligned} \boldsymbol{x}^{\intercal}W= & \begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}=\begin{pmatrix}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu0}}\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu1}}\\ \vdots\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu n}} \end{pmatrix}^{\intercal},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle 0\nu}}=b_{{\scriptscriptstyle \nu}} \end{cases}\\ = & \begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}b_{{\scriptscriptstyle 0}} & b_{{\scriptscriptstyle 1}} & \cdots & b_{{\scriptscriptstyle n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}=\begin{pmatrix}x_{{\scriptscriptstyle i}}w_{{\scriptscriptstyle i0}}+b_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle i}}w_{{\scriptscriptstyle i1}}+b_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{i}w_{{\scriptscriptstyle in}}+b_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal},\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \nu}}=w_{{\scriptscriptstyle 0\nu}} \end{cases} \end{aligned}$$

wrong or incompatible transpose

$$\begin{aligned} \sigma\left(\boldsymbol{x}^{\intercal}W\right)= & \sigma\left(\begin{pmatrix}x_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}w_{{\scriptscriptstyle 00}} & w_{{\scriptscriptstyle 01}} & \cdots & w_{{\scriptscriptstyle 0n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\right)=\sigma\left(\begin{pmatrix}x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu0}}\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu1}}\\ \vdots\\ x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu n}} \end{pmatrix}^{\intercal}\right)=\begin{pmatrix}\sigma_{{\scriptscriptstyle 0}}\left(x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu0}}\right)\\ \sigma_{{\scriptscriptstyle 1}}\left(x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu1}}\right)\\ \vdots\\ \sigma_{{\scriptscriptstyle n}}\left(x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu n}}\right) \end{pmatrix}^{\intercal},\begin{cases} x_{{\scriptscriptstyle 0}}=1\\ w_{{\scriptscriptstyle 0\nu}}=b_{{\scriptscriptstyle \nu}} \end{cases}\\ = & \sigma\left(\begin{pmatrix}1\\ x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle m}} \end{pmatrix}^{\intercal}\begin{pmatrix}b_{{\scriptscriptstyle 0}} & b_{{\scriptscriptstyle 1}} & \cdots & b_{{\scriptscriptstyle n}}\\ w_{{\scriptscriptstyle 10}} & w_{{\scriptscriptstyle 11}} & \cdots & w_{{\scriptscriptstyle 1n}}\\ \vdots & \vdots & \ddots & \vdots\\ w_{{\scriptscriptstyle m0}} & w_{{\scriptscriptstyle m1}} & \cdots & w_{{\scriptscriptstyle mn}} \end{pmatrix}\right)=\sigma\left(\begin{pmatrix}x_{{\scriptscriptstyle i}}w_{{\scriptscriptstyle i0}}+b_{{\scriptscriptstyle 0}}\\ x_{{\scriptscriptstyle i}}w_{{\scriptscriptstyle i1}}+b_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{i}w_{{\scriptscriptstyle in}}+b_{{\scriptscriptstyle n}} \end{pmatrix}^{\intercal}\right)=\sigma_{{\scriptscriptstyle \nu}}\left(x_{{\scriptscriptstyle \mu}}w_{{\scriptscriptstyle \mu\nu}}\right),\begin{cases} 1=x_{{\scriptscriptstyle 0}}\\ b_{{\scriptscriptstyle \nu}}=w_{{\scriptscriptstyle 0\nu}} \end{cases} \end{aligned}$$