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Let $f$ be an endomorphism of a finite dimensional vector space $E$ over the field $K$ and let $P \in K[x]$. Let $P=QR$ with $Q$ and $R$ relatively prime.

Then

$$Ker[P(f)]= Ker[Q(f)]\oplus Ker[R(f)].$$

In French this is called the "lemme des noyaux" or "théorème de la décomposition des noyaux" which translates to kernel decomposition theorem.

Arnaud D.
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math_lover
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  • An answer on https://math.stackexchange.com/questions/275186/the-connection-between-decomposition-of-kerf-phi-g-phi-and-the-chinese-re suggests it doesn't have a common English name. – Patrick Stevens Oct 18 '17 at 07:48
  • That is suprising, as it has many powerful applications in the theory of diagonalisation. – math_lover Oct 18 '17 at 07:55
  • I've discovered the french name when I arrived in France. Before it didn't have a name for me – user84976 Oct 18 '17 at 08:15

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