\begin{align*} Z &= \gamma X \\ &= (I - \frac{1}{N}1 1^T)X \\ &= X - \frac{1}{N} 1 1^T X\\ &= X - 1\frac{1}{N} \begin{bmatrix}\sum^N_{i=1} x_{i1} & \sum^N_{i=1} x_{i2} & \dots & \sum^N_{i=1} x_{id}\end{bmatrix} \\ &= X - 1\begin{bmatrix}\bar{x}_1 & \bar{x}_2 & \dots & \bar{x}_d\end{bmatrix}\\ &= X - 1\bar{x}^T \\ \end{align*}

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Learning from Data

Exercise 9.2

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\begin{array}{l} Z & = γX \\ = (I - \frac{1}{N} 1 1^{T}) X \\ = X - \frac{1}{N} 1 1^{T} X \\ = X - 1 \frac{1}{N} [\begin{matrix} \sum_{i = 1}^{N} x_{i 1} & \sum_{i = 1}^{N} x_{i 2} & \dots & \sum_{i = 1}^{N} x_{id} \end{matrix}] \\ = X - 1 [\begin{matrix} {\bar{x}}_{1} & {\bar{x}}_{2} & \dots & {\bar{x}}_{d} \end{matrix}] \\ = X - 1 {\bar{x}}^{T} \end{array}

niuers

2021-12-08 10:20

Exercise 9.2

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