Problem 1 [35 points] Symmetric positive (semi)definite matrices

(a) [(1+4) + (1+4) = 10 points] Outer product matrix

For $x\in\mathbb{R}^{n}\setminus\{0\}$, consider the matrix $X = x x^{\top}$.

(i) Is $X\in\mathbb{S}_{+}^{n}$? Explain why/why not.

(ii) Is $X\in\mathbb{S}_{++}^{n}$? Explain why/why not.

(b) [(1+4) + (1+4) = 10 points] Copositive matrix

Let $\mathcal{X}$ denote the set of $n\times n$ real copositive matrices (see Lec. 3, p. 1-2).

(i) True or false: $\mathcal{X} \subset \mathbb{S}_{+}^{n}$. Explain your answer.

(ii) True or false: $\mathbb{S}_{+}^{n} \subset \mathcal{X}$. Explain your answer.

(c) [15 points] Visualizing $\mathbb{S}_{+}^{2}$

Consider $X = \begin{pmatrix} x & z\\ z & y \end{pmatrix} \in \mathbb{S}_{+}^{2}$ where $x,y,z$ are scalars. Discretize $x,y$ within appropriate intervals and use your favorite programs such as MATLAB/Python/Julia to visualize $\mathbb{S}_{+}^{2}$ as a subset of $\mathbb{R}^{3}$, that is, plot it as a three dimensional set.

Insert the plot in the notebook. Submit your code in the zip file so that we can reproduce your plot.

(Hint: consider the principal minor characterization from Lec. 3, p. 18-20)

Problem 2 [5 x 6 = 30 points] Vector unit norm balls

For any fixed $p$ satisfying $0\leq p \leq \infty$, the vector unit $p$-norm ball is a set

$$ \{x\in\mathbb{R}^{n} : \|x\|_{p} \leq 1\} \subset \mathbb{R}^{n}.$$

Clealry, the above set is centered at the origin. For the definition of vector $p$-norm, see Lec. 3, p. 5 .

The following plot shows the two dimensional $p$-norm balls for $p\in\{0.5,1,1.5,2,3.5,\infty\}$ (from left to right, top to bottom).

Use your favorite programs such as MATLAB/Python/Julia to plot the three dimensional $p$-norm balls for the same $p$ as above. Insert the plot in the notebook. Submit your code in the zip file so that we can reproduce your plot.

Problem 3 [35 points] Schatten $p$-norm of a matrix

In Lec. 3, p. 6-8, we discussed the induced $p$-norm of any matrix $X\in\mathbb{R}^{m\times n}$. A different way to define matrix norm is to simply consider the $p$-norm of the vector comprising of the singular values of $X$.

Specifically, the Schatten $p$-norm of a matrix $X\in\mathbb{R}^{m\times n}$ is

$$\|X\|_{\text{Schatten}\;p} := \left(\displaystyle\sum_{i=1}^{\min\{m,n\}}\left(\sigma_{i}(X)\right)^{p}\right)^{1/p}.$$

In other words, if we define a vector $\sigma := (\sigma_1, ..., \sigma_{\min\{m,n\}})$, then $\|X\|_{\text{Schatten}\;p} = \|\sigma\|_{p}$.

(a) [10 points] Schatten 2-norm

Prove that $\|X\|_{\text{Schatten}\;2} = \|X\|_{\text{F}}$, the Frobenius norm (see Lec. 3, p. 9 bottom).

(b) [10 points] Schatten $\infty$-norm

Prove that $\|X\|_{\text{Schatten}\;\infty} = \|X\|_{\text{Induced}\;2}$, the spectral norm (see Lec. 3, p. 8 bottom).

(c) [10 points] Schatten 1-norm

Prove that if $X\in\mathbb{S}^{n}_{+}$, then $\|X\|_{\text{Schatten}\;1} = \text{trace}(X)$.

Remark: Schatten 1-norm is also called the nuclear norm, and is extremely important in convex optimization. We will learn more about it later in this course.

(d) [1+4 = 5 points] Schatten 0-norm

The vector $0$-norm is defined as the cardinality (that is, number of nonzero entries) of the vector.

What is the interpretation of Schatten 0-norm $\|\sigma\|_{0}$? Explain your answer.