# Trace of a power of a skew-symmetric matrix

How to express $${\rm Tr}(A^n)$$ (in terms of $${\rm det}\,A$$), where $$A$$ is a skew-symmetric $$m\times m$$ matrix? With references if possible.

For even powers we cannot determine the trace from the determinant and order alone.

Let matrix $$A$$ be the $$3×3$$ matrix defined as follows:

$$A_{1,2}=A_{2,3}=A_{3,1}=c,$$

with other entries defined by skew-symmetry. Then $$\det A=0$$. But $$A^2$$ has all its diagonal elements equal to $$-2c^2$$, so its trace is $$-6c^2$$ which varies with $$c$$ despite the constant determinant.

The trace of an odd power of any skew-symmetric matrix is always zero. Can you see why?

• Sorry, ignore that :)
– Kosm
May 5 at 10:43
• On the vanishing of the trace for odd power, could you explain why? Not sure how to prove it
– Kosm
May 5 at 11:13
• $(A^n)^T=-A^n$ for odd $n$, so $A^n$ is also skew-symmetric. May 5 at 11:21