Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
2026-03-30 13:38:43.1774877923
Eigenvalues of a $3\times3$ orthogonal matrix
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Example. Let $$ U=\left(\begin{matrix}1&0&0\\ 0&0&1\\0&-1&0 \end{matrix}\right). $$ Then, $U$ is orthogonal and its characteristic polynomial is $$ p(\lambda)=\det\,\left(\begin{matrix}1-\lambda&0&0\\ 0&-\lambda&1\\0&-1&-\lambda \end{matrix}\right) =-(\lambda-1)(\lambda^2+1). $$