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.
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.
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)
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.
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 $\|X\|_{\text{Schatten}\;p} = \|\sigma\|_{p}$.
Prove that $\|X\|_{\text{Schatten}\;2} = \|X\|_{\text{F}}$, the Frobenius norm (see Lec. 3, p. 9 bottom).
Prove that $\|X\|_{\text{Schatten}\;\infty} = \|X\|_{\text{Induced}\;2}$, the spectral norm (see Lec. 3, p. 8 bottom).
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.
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.