[Math] Is every skew-symmetric matrix congruent to a diagonal matrix

linear algebramatrix-congruences

Question

Prove/disprove: if A, a matrix nxn over field F is skew-symmetric then A congruents with a diagonal matrix.

My thoughts

I know that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. But is it true for skew-symmetric as well?
I really have no idea what to do here…
I know A consists of zeros on it's diagonal. it means it's nilpotent? if so, the eigenvalues are only zeros, I guess it means that its diagonal matrix should be all zeros?

I'm really confused here,
any lead would help,
many thanks.

Best Answer

Hint. Suppose $\operatorname{char}(\mathbb F)\ne2$. If $A$ is skew-symmetric and $A=P^TDP$ for some invertible matrix $P$ and diagonal matrix $D$, then $D$ has to be skew-symmetric too.

When $\operatorname{char}(\mathbb F)=2$, consider $A=\pmatrix{0&1\\ 1&0}$ and also $A=I_2$.

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