A matrix that:
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Is not diagonalizable over \({\mathbf{R}}\) but diagonalizable over \({\mathbf{C}}\)
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Has no eigenvalues over \({\mathbf{R}}\) but has distinct eigenvalues over \({\mathbf{C}}\)
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\(\min_M(x) = \chi_M(x) = x^2 + 1\)
\begin{align*} M = \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right) \sim \left(\begin{array}{r|r} -1 \sqrt{-1} & 0 \\ \hline 0 & 1 \sqrt{-1} \end{array} \right) .\end{align*}
A matrix that:
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Is not diagonalizable over \({\mathbf{C}}\),
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Has eigenvalues \([1, 1]\) (repeated, multiplicity 2)
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\(\min_M(x) = \chi_M(x) = x^2-2x+1\)
\begin{align*} M = \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \sim \left( \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right) .\end{align*}
Non-similar matrices with the same characteristic polynomial \begin{align*} \left(\begin{array}{ll} {0} & {0} \\ {0} & {0} \end{array}\right) \text { and } \left(\begin{array}{ll} {0} & {0} \\ {0} & {0} \end{array}\right) \end{align*} Here \(\chi_A(x) = \chi_B(x) = x^2\), but they are not conjugate since their JCFs differ (note that they’re already in JCF!)
A full-rank matrix that is not diagonalizable: \begin{align*} \left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right) .\end{align*}
Matrix roots of unity, i.e. representations of \(i\): \begin{align*} M_1 \coloneqq { \begin{bmatrix} {0} & {-1} \\ {1} & {0} \end{bmatrix} } \quad M_2 \coloneqq { \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} } .\end{align*}