Homework 2

Due: Oct. 15, 14:00:00

Definite matrices

1. Suppose $\mathbf{A}\in {\mathbb{R}}^{n\times n}$$\mathbf{A}\in \mathbb{R}^{n\times n}$ is a real symmetric matrix. Show that ${\mathbf{A}}^{2k}$$\mathbf{A}^{2k}$ is positive semidefinite for all $k=1,2,3,\dots$$k = 1, 2, 3, \ldots$.

2. Let $\begin{array}{c}\mathbf{A}=\left(\begin{array}{ccc}2& 2& a\\ 2& 4& -2\\ a& -2& 2\end{array}\right)\end{array}$$\mathbf{A} = \begin{pmatrix}2 & 2 & a\\2 & 4 & -2\\a & -2 & 2\end{pmatrix}$.

   • If $a=-1$$a=-1$, compute the eigenvalues of $\mathbf{A}$$\mathbf{A}$ and show that $\mathbf{A}$$\mathbf{A}$ is positive definite.
   • If $a=-2$$a=-2$, use Sylvester's criterion to show that $\mathbf{A}$$\mathbf{A}$ is positive semidefinite.
   • For what real numbers $a$$a$ is matrix $\mathbf{A}$$\mathbf{A}$ positive semidefinite? You can use any criterion you like.

Convex sets

3. Show that if $C_1,C_2\subseteq {\mathbb{R}}^n$$C_1, C_2\subseteq \mathbb{R}^n$ are two convex sets, so is $C_1+C_2\triangleq \{x_1+x_2:x_1\in C_1,x_2\in C_2\}$$C_1 + C_2 \triangleq \{x_1 + x_2 : x_1 \in C_1, x_2\in C_2\}$.
4. Let $f(\mathbf{x})=\mathbf{Ax}\mathbf{+}\mathbf{b}$$f(\mathbf{x}) = \mathbf{Ax+b}$ be an affine function from ${\mathbb{R}}^n$$\mathbb{R}^n$ to ${\mathbb{R}}^m$$\mathbb{R}^m$, where $\mathbf{A}\in {\mathbb{R}}^{m\times n}$$\mathbf{A} \in \mathbb{R}^{m\times n}$ and $\mathbf{b}\in {\mathbb{R}}^m$$\mathbf{b}\in \mathbb{R}^m$. Show that if $C\subseteq {\mathbb{R}}^m$$C\subseteq \mathbb{R}^m$ is a convex set, so is its inverse image $f^{-1}(C)\triangleq \{\mathbf{x}:f(\mathbf{x})\in C\}$$f^{-1}(C)\triangleq \{\mathbf{x}: f(\mathbf{x}) \in C\}$.