Exercise 1.8.11

The trace of $S$ is $0$. By equation $(21)$, its eigenvalues are in pairs of ${\sigma }_k$ and $-{\sigma }_k$, so the sum of eigenvalues are also $0$.

If $A$ is a square diagonal matrix with entries $1,2,\dots ,n$, then its eigenvalues are $1,2,\dots ,n$. Since its real, symmetric matrix, Both its right and left singular vectors are $x_i=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \end{array}\right]$, where $1$ appears in the $i$th position for the $i$th eigenvalue.

The $2n$ eigenvalues of $S$ are: $1,2,\dots ,n,-1,-2,\dots ,-n$.

The $2n$ eigenvectors of $S$ are: $s_i=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \end{array}\right]$ and $s_j=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill -1\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill \end{array}\right]$

Where $i,j=1,2,\dots ,n$