If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?

Question

This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?

The answer is yes. One way to see this is to consider the trace of $A^2$ where $A$ is the skew-symmetric matrix in question. — user1551, May 17 '22 at 09:04

score 0 · Answer 1 · answered May 17 '22 at 08:48

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If the matrix has real entries: Any matrix with all eigenvalues zero is nilpotent, the only nilpotent skew-symmetric matrix is the zero matrix.

answered May 17 '22 at 08:48

G. Gare

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Begs the question, why the only nilpotent skew-symmetric matrix is the zero matrix. – Gerry Myerson May 17 '22 at 08:54
$A^2$ is symmetric – G. Gare May 17 '22 at 09:01
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So now the question is why the only nilpotent symmetric matrix is the zero matrix. But of course symmetric matrices are diagonalizable. – Gerry Myerson May 17 '22 at 09:12
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https://math.stackexchange.com/questions/478470/is-the-zero-matrix-the-only-symmetric-nilpotent-matrix-with-real-values – G. Gare May 17 '22 at 10:39

1 Answers1