## Three by three skew-symmetric matrix

This is just a short remark about 3 x 3 skew-symmetric-matrix.

Theorem.          If $X\in \mathbb{R}^{3\times 3}$ is nonzero and skew-symmetric, then two of its singular values are nonzero and equal, while the other one is zero.

Proof.          If $X\in \mathbb{R}^{3\times 3}$ is nonzero and skew-symmetric, then $X$ is normal, so its singular values are the moduli of its eigenvalues. Moreover the eigenvalues are purely imaginary. Since the characteristic polynomial of $X$ is a univariate cubic with real coefficients,  it has either one real root only or three real roots. If it has three real roots then they must be all zero. So $X = 0$, violating the assumption. Hence $X$ has one real root which is zero, and two nonzero purely imaginary roots which are conjugate to each other. Then these two imaginary roots have the same modulus. Therefore $X$ has two equal nonzero singular values while the other one is zero. $\square$

So how to describe all complex 3 by 3 matrices that have the properties mentioned in the above result? If having time we may add a conjecture later.

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