**Unformatted text preview: **ordan form; i.e., prove that every
A ∈ C n×n satisﬁes its own characteristic equation. It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] 7.8.7. Prove that if λ is an eigenvalue of A ∈ C n×n such that index (λ) = k
and alg multA (λ) = m, then dim N (A − λI)k = m. Is it also true
that dim N (A − λI)m = m?
7.8.8. Let λj be an eigenvalue of A with index (λj ) = kj . Prove that if
Mi (λj ) = R (A − λj I)i ∩ N (A − λj I), then D
E 0 = Mkj (λj ) ⊆ Mkj−1 (λj ) ⊆ · · · ⊆ M0 (λj ) = N (A − λj I). T
H 7.8.9. Explain why (A − λj I)i x = b(λj ) must be a consistent system whenever
λj ∈ σ (A) and b(λj ) ∈ Si (λj ), where b(λj ) and Si (λj ) are as deﬁned
on p. 594. IG
R 7.8.10. Does the result of Exercise 7.7.5 extend to nonnilpotent matrices? That
is, if λ ∈ σ (A) with index (λ) = k > 1, is Mk−1 = R (A − λI)k−1 ?
7.8.11. As deﬁned in Exercise 5.8.15 (p. 380) and mentioned in Exercise 7.6.10
80
(p. 573), the Kronecker product (sometimes called tensor product ,
80 Y
P Leopold Kronecker (1823–1891) was born in Liegnitz, Prussia (now Legnica, Poland), to a
wealthy business family that hired private tutors to educate him until he enrolled at Gymnasium at Liegnitz where his mathematical talents were recognized by Eduard Kummer (1810–
1893), who became his mentor and lifelong colleague. Kronecker went to Berlin University
in 1841 to earn his doctorate, writing on algebraic number theory, under the supervision of
Dirichlet (p. 563). Rather than pursuing a standard academic career, Kronecker returned to
Liegnitz to marry his cousin and become involved in his uncle’s banking business. But he never
lost his enjoyment of mathematics. After estate and business interests were left to others in
1855, Kronecker joined Kummer in Berlin who had just arrived to occupy the position vacated
by Dirichlet’s move to G¨ttingen. Kronecker didn’t need a salary, so he...

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