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We know that when the characteristic polynomial of a linear operator $f$ can be written in the form: $$ \chi_f = \prod_i (x - \lambda_i)^{\mu_i}, $$ the number $\mu_i$ is called the algebraic multiplicity of the eigenvalue $\lambda_i$. My question is: when the minimal polynomial of a linear operator $f$ can be written in the form: $$ \pi_f = \prod_i (x - \lambda_i)^{d_i}, $$ what do we call the number $d_i$ associated to the eigenvalue $\lambda_i$? Thanks!

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    $\begingroup$ I don’t believe it has a name. It corresponds to the size of the largest Jordan block associated to that eigenvalue. If I had to make up a name I might go with “width” or “height.” $\endgroup$ – Qiaochu Yuan Nov 21 '20 at 19:25
  • $\begingroup$ There is the "geometric multiplicity", which is the dimension of the associated eigenspace (the algebraic multiplicity is the dimension of the associated generalized eigenspace). However, I do not recall off-hand if it corresponds to the multiplicity of the eigenvector as a root of the minimal polynomial. $\endgroup$ – Paul Sinclair Nov 22 '20 at 1:02
  • $\begingroup$ A stab at a module theoretic-definition of $d_i$ would be "the minimal length of a filtration of the $\lambda_i$-block by a filtration with semisimple quotients". Unfortunately the "length" of a module is the length of a filtration with simple quotients, which turns out to be $\mu_i$ for the $\lambda_i$-block, and I don't know if the semisimple version has a specific name. $\endgroup$ – Joppy Nov 23 '20 at 12:19

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