Homework 3

Problem 1

Consider the matrix

$$\pmb{A} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$$

1. Show that $\pmb{A}$ is positive definite.

2. Find $\pmb{A}^{1/2}$.

Problem 2

Let

$$\pmb{A} = \begin{pmatrix} 1 & 0 & x \\ 0 & 1 & 0 \\ x & 0 & 1 \\ \end{pmatrix}$$

Find $\frac{\partial \log\vert\pmb{A}\vert}{\partial x}$

1. directly

2. using theorem (4.3.30)

Problem 3

Say $\pmb{Y} = \begin{pmatrix} Y_1& Y_2 \end{pmatrix}'$ with $E[Y_1]=E[Y_2]=0$, $var(Y_1)=1$, $var(Y_2)=4$ and $cov(Y_1,Y_2)=-2$. Let $\pmb{A} = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}$.

Verify theorem (5.1.10) part i. directly.