# Chapter 1 Preliminaries

• Idempotent: A matrix $$A$$ is idempotent if $$AA = A$$

• The rank of an idempotent matrix is equal to its trace, i.e. $$rank(A) = trace(A)$$
• Generalized Inverse: $$G$$ is a generalized inverse of $$A$$ if $$AGA = A$$

• Quadratic form, positive definite, non-negative definite

• $$A$$ is PD $$\Rightarrow$$ $$A^{-1}$$ exists
• linear transformation

• multivariate normal distribution

• if $$y \sim N(\mu, \Sigma)$$, then $$y \stackrel{d}{=} Az + \mu$$ where $$z\sim N(0, I_{n\times n})$$ and $$A$$ is an $$m\times n$$ matrix of rank $$n$$ such that $$AA' = \Sigma$$

• if $$y\sim N(\mu, I_{n\times n})$$, then $$w = y'y \sim \chi_n^2(\mu'\mu/2)$$

• Suppose $$\Sigma$$ is an $$n\times n$$ positive definite matrix, $$A$$ is an $$n\times n$$ symmetric matrix of rank $$m$$ such that $$A\Sigma$$ is idempotent ($$A\Sigma A\Sigma = A\Sigma$$). Then $y \sim N(\mu, \Sigma) \Rightarrow y'Ay\sim \chi_m^2(\mu'A\mu/2)$

• If $$w\sim \chi_m^2(\theta)$$ then $$E(w) = m + 2\theta$$ and $$Var(w) = 2m + 8\theta$$

• Non-central t distribution: $$y\sim N(\delta, 1)$$, $$w \sim \chi_m^2$$, $$y$$ and $$w$$ are independent then $$\frac{y}{\sqrt{w/m}}$$ has a non-central $$t$$ distribution with $$m$$ degrees of freedom and non-centrality parameter $$\delta$$

• Non-central F distribution: $$w_1 \sim \chi_{m_1}^2(\theta)$$ and $$w_2\sim \chi_{m_2}^2$$. $$w_1$$ and $$w_2$$ are independent then $$\frac{w_1/m_1}{w_2/m_2}$$ has a non-central F distribution with $$m_1$$ numerator degrees of freedom, $$m_2$$ denominator degrees of freedom and non-centrality parameter $$\theta$$: $$\frac{w_1/m_1}{w_2/m_2} \sim F_{m_1, m_2}(\theta)$$.

• if $$u\sim t_m(\delta)$$, then $$u^2 \sim F_{1, m}(\delta^2/2)$$.