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Let $M$ be a $n$ by $n$ matrix over a field $F$.

  1. $M$ is diagonizable, i.e. $M=P DP^{-1}$ for some invertible matrix $P$ and some diagonal matrix $D$, if and only if there exists an eigenbasis.

    I wonder if $M$ can be similar to some special matrix if and only if there exists a generalized eigenbasis. Note a generalized eigenvector $v$ for an eigenvalue $\lambda$ with algebraic multiplicity $c$ is defined as a vector which satisfies $ (M - \lambda I)^c v = 0$.

  2. $M$ admits a Jordan decomposition $M=P JP^{-1}$ for some invertible matrix $P$ and some Jordan canonical form $J$, if and only if the characteristic polynomial of $M$ can split into linear factors over $F$.

    Since columns of $P$ form a generalized eigenbasis, "the characteristic polynomial of $M$ can split into linear factors over $F$" implies "there exists a generalized eigenbasis", but I wonder if the reverse is false, i.e. "there exists a generalized eigenbasis" doesn't necessarily imply "the characteristic polynomial of $M$ can split into linear factors over $F$"?

Thanks!

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Having a generalized eigenbasis is equivalent to the existence of Jordan decomposition.

One way to see this is by noting that $\prod_{\lambda}(M-\lambda I)^{c(\lambda)}=0$ because the image of each vector in the generalized eigenbasis is 0 under this function.

Now, the minimal polynomial of $M$ divides this, so it splits in linear factors, but the characteristic polynomial shares all irreducible factors with the minimal polynomial (albeit with possibly different multiplicities), so the characteristic polynomial also splits into linear factors.

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