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.

Advertisements
This entry was posted in Linear Algebra and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s