確率変数多次元ベクトル
\(X_i, i \in \mathbb{N} _+,i \leq n \)となるindex付きの確率変数を用意して, \[ \boldsymbol{X} = \begin{pmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{pmatrix} \] この,平均を \(\mathrm{E}[X_i] = \mu_i\) と分散,共分散を\( \sigma_{ij} = \mathrm{Cov}(X_i,X_j) \)として, \[ \mathrm{E}[\boldsymbol{X}] = \boldsymbol{\mu} = \begin{pmatrix} \mu_1 \\ \mu_2 \\ \vdots \\ \mu_n \end{pmatrix} \\ \mathrm{V}[\boldsymbol{X}] = \boldsymbol{\Sigma} = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1} & \sigma_{n2} & \cdots & \sigma_{nn} \end{pmatrix} \\ \] 表す. \(\mathbf{A} \in \mathbb{R}^{m \times n},\boldsymbol{b} \in \mathbb{R}^{n}\)として, \[ \boldsymbol{Z} = \mathbf{A}\boldsymbol{X} + \boldsymbol{\mu} \] という線形変換を行ったとき, \[ \begin{align} \mathrm{E}[\boldsymbol{Z}] &= \mathrm{E}[\mathbf{A}\boldsymbol{X} + \boldsymbol{\mu}] \\ &=\mathrm{E}[\mathbf{A}\boldsymbol{X}] + \boldsymbol{\mu} \\ &= \mathbf{A}\mathrm{E}[\boldsymbol{X}] + \boldsymbol{\mu} \end{align} \] \[ \begin{align} \mathrm{V}[\boldsymbol{Z}] &= \mathrm{V}[\mathbf{A}\boldsymbol{X} + \boldsymbol{\mu}] \\ &= \mathrm{V}[\mathbf{A}\boldsymbol{X}] \\ &=\mathrm{E}[(\mathbf{A}\boldsymbol{X} - \mathrm{E}[\mathbf{A}\boldsymbol{X}])(\mathbf{A}\boldsymbol{X} - \mathrm{E}[\mathbf{A}\boldsymbol{X}])^{\mathsf{T}}] \\ &= \mathrm{E}[\mathbf{A}(\boldsymbol{X} - \mathrm{E}[\boldsymbol{X}])(\boldsymbol{X} - \mathrm{E}[\boldsymbol{X}])^{\mathsf{T}}\mathbf{A}^{\mathsf{T}}] \\ &= \mathbf{A}\mathrm{E}[(\boldsymbol{X} - \mathrm{E}[\boldsymbol{X}])(\boldsymbol{X} - \mathrm{E}[\boldsymbol{X}])^{\mathsf{T}}]\mathbf{A}^{\mathsf{T}} \\ &= \mathbf{A}\mathbf{\Sigma}\mathbf{A}^{\mathsf{T}} \\ \end{align} \] となる.