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Suppose $A$ is a real skew-symmetric matrix, and $\tilde{A}$ is an induced symmetric matrix created by flipping the sign of all elements below the diagonal. As we know, the spectrum of $A$ is purely imaginary and consists of pairs (if $3i$ is an eigenvalue, then $-3i$ must also be an eigenvalue), but the spectrum of $\tilde{A}$ may contain real numbers all distinct from each other.

I wonder if there is any relationship between the spectrum of $A$ and $\tilde{A}$. After all, they can be fully recovered from each other in a trivial way (assuming the diagonal of $\tilde{A}$ is all zero), so why does one give a paired spectrum whereas the other may have a completely heterogeneous spectrum?

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  • $\begingroup$ (This doesn't answer the question, but) both $A$ and $\tilde A$ can be gotten from a strictly upper triangular matrix $U$: $$A=U-U^T,\qquad\tilde A=U+U^T$$ And the spectrum of $U$ is trivial, $\{0\}$. $\endgroup$
    – mr_e_man
    Dec 5, 2022 at 20:14

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Let $K$ be a skew symmetric matrix and $S$ be its so-called induced symmetric counterpart.

When $K$ is $3\times3$, the eigenvalues of $K$ are $ik,-ik$ and $0$ for some $k\in\mathbb R$, while the eigenvalues of $S$ are some $a,b,c\in\mathbb R$. There are only two things that we can say about the relationship between $a,b,c$ and $k$, namely,

  1. $a+b+c=0$, i.e. $\operatorname{tr}(K)=\operatorname{tr}(S)=0$, because both matrices have zero diagonals;
  2. $a^2+b^2+c^2=2k^2$, i.e., $K$ and $S$ have the same Frobenius norm.

Nothing further can be said without additional information. You wrote:

After all, they can be fully recovered from each other in a trivial way (assuming the diagonal of $\tilde{A}$ is all zero),

but this relationship is only superficial, because it is not preserved under orthogonal similarity. That is, if $S$ is induced from $K$ and $S_1$ is induced from $QKQ^T$ for some orthogonal matrix $Q$, then $S$ and $S_1$ in general are not even similar.

In fact, given any four real numbers $k,a,b,c$ satisfying $a^2+b^2+c^2=2k^2$ and $a+b+c=0$, there always exists a skew symmetric matrix $K$ such that its spectrum is $\{ik,-ik,0\}$ and the spectrum of its induced symmetric matrix $S$ is $\{a,b,c\}$. This is because every traceless diagonal matrix is orthogonally similar to a symmetric matrix with a zero diagonal and the spectrum of a $3\times3$ skew-symmetric matrix is uniquely determined by its Frobenius norm.

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