Proposition 11

Tag 076F, permanent. Cite it as [Bourbaki, Tag 076F].

Algebra, chapter VII, § 5. Endomorphisms of vector spaces

Let E (resp. $E'$ ) be a finite dimensional vector space over a commutative field K and let u (resp. $u'$ ) be an endomorphism of E (resp. $E'$ ). Let*

\chi_u(X) = \prod_i (X - \alpha_i), \quad \chi_{u'}(X) = \prod_j (X - \beta_j)

be decompositions into linear factors of the characteristic polynomials of u and $u'$ in some suitable extension of $K$ . Then the characteristic polynomial of the endomorphism $u \otimes u'$ of the vector space $E \otimes_K E'$ is given by the formula

\chi_u \otimes u'(X) = \prod_{i,j} (X - \alpha_i \beta_j).

Arguing as in the proof of Prop. 10 of VII, p. 36, we see that it is sufficient to prove the following lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Prop. 10