I want to see what the unit nuclear norm ball looks like.
So I think of matrices whose singular values add up to $1$. For simplicity, let's talk about symmetric, $2\times 2$ matrices (so that I can limit myself to $3$ dimensions). These matrices can be thought of as points in a $3$-dimensional space, and the coordinate values tell us about the entries in the matrix. But what shape will the matrices that have nuclear norm $1$ form?
I have seen figures showing it to be like a solid cylinder, but I can't see why.
I have considered a symmetric matrix $$ A=\left( \begin{array}{cc} a & b\\ b &c\end{array} \right)$$ where $a, b, c$ are real numbers. I calculated the singular values $s_{1,2}$ of $A$ as positive square roots of the eigenvalues of $A^2$. If I have not make a mistake, then $$ s_{1,2}=\frac{1}{\sqrt{2}}\sqrt{a^2+2b^2+c^2\pm|a+c|\sqrt{(a-c)^2+4b^2}}. $$ Now we want that $s_1+s_2=1$. After squaring I received $$ a^2+2b^2+c^2+\sqrt{(a^2+2b^2+c^2)^2-(a+c)^2((a-c)^2+4b^2)}=1$$ and consequently $$ a^2+2b^2+c^2+2|b^2-ac|=1. $$ Thus, if $ac\geq b^2$, then $$(a+c)^2=1,$$ and if $ac<b^2$, then $$ (a-c)^2+4b^2=1.$$
I hope that I have not make a mistake in my calculations.