# Relationship between the spectrum of a skew-symmetric matrix and its symmetric counterpart

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?

• (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, $\{0\}$. Dec 5, 2022 at 20:14

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.