Homework 2

Deadline: Oct. 14, 22:00:00

Definite matrices

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

2. Let $\mathbf{A} = \begin{pmatrix}2 & 2 & a\\2 & 4 & -2\\a & -2 & 2\end{pmatrix}$.

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

Convex sets

3. Show that an ellipsoid $\{x=(x_1,x_2,\ldots, x_n)^{\mathsf{T}}\in \mathbb{R}^n :\frac{x_1^2}{\lambda_1^2}+\frac{x_2^2}{\lambda_2^2}+\cdots +\frac{x_n^2}{\lambda_n^2}\leq 1\}$ is the image of a ball under a linear transformation.
4. Verify that the Minkowski additionis a convexity-preserving operation. Namely, show that if $C_1, C_2\subseteq \mathbb{R}^n$ are two convex sets, so is $C_1 + C_2 \triangleq \{x_1 + x_2 : x_1 \in C_1, x_2\in C_2\}$.
5. Let $f(\mathbf{x}) = \mathbf{Ax+b}$ be an affine function from $\mathbb{R}^n$ to $\mathbb{R}^m$, where $\mathbf{A} \in \mathbb{R}^{m\times n}$ and $\mathbf{b}\in \mathbb{R}^m$. Show that if $C\subseteq \mathbb{R}^m$ is a convex set, so is its inverse image $f^{-1}(C)\triangleq \{\mathbf{x}: f(\mathbf{x}) \in C\}$.