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Are two matrices similar if and only if they have the same Jordan Canonical form?

Does the Jordan form have to have ordered eigenvalues?

For example, if $\lambda_1$ and $\lambda_2$ are eigenvalues of $A$, are $\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}$ and $\begin{pmatrix}\lambda_2&0\\0&\lambda_1\end{pmatrix}$ both Jordan forms of $A$?

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  • $\begingroup$ The Jordan normal form exists for complex matrices (or more generally matrices over an algebraically closed field). The eigenvalues of a complex matrix are complex numbers, and there is no natural "order" on $\mathbb C$, so your second question doesn't really make sense. $\endgroup$
    – krm2233
    Commented Apr 12, 2023 at 18:04
  • $\begingroup$ You can order complex numbers, if it's just about order and addition doesn't matter. Put a+bi in a tuple (a,b) and order it. So one could agree on a "canonical" Jordan form for convenience. $\endgroup$
    – Gere
    Commented Apr 25 at 9:35

2 Answers 2

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  • Up to arbitrary ordering of Jordan blocks, yes
  • No
  • Yes
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  • $\begingroup$ Thanks! That really cleared my problem up! $\endgroup$ Commented Nov 11, 2014 at 22:30
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The eigenvalues need not be ordered. You can conjugate a diagonal matrix by a permutation matrix to put them in any order.

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