Transformation

Theorem)
Let $X \sim N_n(\mu,\Sigma)$ with $\mu \in \mathbb{R}^{n \times n}$,$\Sigma:$ n$\times$n positive definite matrix.
Then, $Y=AX+b \sim N_m(A\mu+b, A \Sigma A^T)$
pf)
$M_Y(t)=E(exp(t^TY))=exp(t^Tb)E(exp(t^TAX))=exp(t^Tb)exp(t^TA\mu+\frac{1}{2} t^T A \Sigma A^T t)$: mgf of $N_m(A \mu + b, A \Sigma A^T)$

Corallary)
Suppose $X \sim N_n(\mu,\Sigma)$ and $X= \begin{pmatrix} X_1\\ X_2 \end{pmatrix}$, $\mu= \begin{pmatrix} \mu_1\\ \mu_2 \end{pmatrix}$, $\Sigma= \begin{pmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ where $X_1, \mu_1 \in \mathbb{R}^m$ and $\Sigma_{11} \in \mathbb{R}^{m \times m}(m < n)$
Then, $X_1 \sim N_m(\mu_1,\Sigma_{11})$ and $X_2 \sim N_{n-m}(\mu_2,\Sigma_{22})$
$\because$
$X_1=AX+b$ where $A= \begin{pmatrix} I_m & 0_{m \times (n-m)} \end{pmatrix}$, $b=0$
$X_2=AX+b$ where $A= \begin{pmatrix} 0_m & I_{m \times (n-m)} \end{pmatrix}$, $b=0$

Independence

Theorem)
Suppose $X \sim N_n(\mu, \Sigma)$ and $X= \begin{pmatrix} X_1\\ X_2 \end{pmatrix}$,$\mu= \begin{pmatrix} \mu_1\\ \mu_2 \end{pmatrix}$, $\Sigma= \begin{pmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ where $X_1, \mu_1 \in \mathbb{R}^m$ and $\Sigma_{11} \in \mathbb{R}^{m \times m}(m < n)$
Then, $X_1 \in \mathbb{R}^m$ and $X_2 \in \mathbb{R}^{n-m}$ are indep. <=> $\Sigma_{12}=0$
pf)

Corallary)
If $X \sim N_n(\mu,\Sigma)$, then $AX \perp\!\!\!\perp BX <=> A \Sigma B^T =0$
$\because$
Note that $\begin{pmatrix} AX \\ BX \end{pmatrix}=\begin{pmatrix} A \\ B \end{pmatrix}X \sim Normal$
$AX \perp\!\!\!\perp BX <=> Cov(AX,BX)=A \Sigma B^T=0$

Conditional distribution

Theorem)
Suppose $X \sim N_n(\mu, \Sigma)$ and $X= \begin{pmatrix} X_1\\ X_2 \end{pmatrix}$,$\mu= \begin{pmatrix} \mu_1\\ \mu_2 \end{pmatrix}$, $\Sigma= \begin{pmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ where $X_1, \mu_1 \in \mathbb{R}^m$ and $\Sigma_{11} \in \mathbb{R}^{m \times m}(m < n)$
Then, if $\Sigma_{22}$ is positive definite, $X_1|X_2=x_2 \sim N_m(\mu_1 + \Sigma_{12} \Sigma_{22}^{-1}(x_2 - \mu_2), \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21})$
pf)

Relationship with $\chi^2$ distribution

Theorem)
Let $X \sim N_n(\mu, \Sigma)$ with $\mu \in \mathbb{R}^n$, $\Sigma$: $n \times n$ positive definite matrix
Then, $Y=(X-\mu)^T \Sigma^{-1} (X-\mu) \sim \chi^2_n$
$\because$
Since $X \overset{d}{\equiv} \Sigma^{1/2}Z+\mu,Y=(X-\mu)^T \Sigma^{-1/2} \Sigma^{-1/2} (X-\mu)=Z^T Z=\sum_i Z_i^2 \sim \chi_n^2$

Theorem) Distribution of quadratic form
Suppose $Z \sim N_n(0,I_n)$ and $A \in \mathbb{R}^{n \times n}$ is symmetric
If $A^2=A$ with $tr(A)=r$, then $Z^T A Z \sim \chi^2_r$
pf)