It seems $X^2+X+1$ is the minimal polynomial of $X$, but it does not have real root, so does this matrix has real Jordan form?
2026-03-29 10:19:25.1774779565
Jordan forms of a real 3×3 matrix X satisfying $X^2+X+I=0$.
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Such a matrix does have a Jordan normal form with complex entries. Because it's a real matrix, its complex Jordan blocks come in conjugate pairs. In fact, this Jordan form must also be diagonal.
It also has a real normal form.