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Matrix Analysis 4 / 34Matrix Analysis Dept Math
5. λ1, λ2, · · · , λs ´
A p Ø Ó A §
´ é A A 5 Ã ' A
xii1, xii2, · · · , xiririi λii
þ (i = 1, 2, · · · , s)§ K þ |
x11, · · · , x1r1 , x21, · · · , x2r2 , · · · , xs1, · · · , xsrss
5 Ã ' "
Nanjing University of Science and Technology 5 / 34Matrix Analysis Dept Math
6. f (λ) ´ λ õ ª
s s−1
f (λ) = asλ + as−1λ + · · · + a1λ + a0
é u A ∈ C n×n § 5 ½
s s−1
f (A) = asA + as−1A + · · · + a1A + a0I
¡ f (A) Ý
A õ ª "
Nanjing University of Science and Technology 6 / 34Matrix Analysis Dept Math
7. A n A λ1, λ2, · · · , λn § é A A
þ x1, x2, · · · , xn § f (λ) õ ª § K f (A)
A f (λ1), f (λ2), · · · , f (λn)§ é A A
þ E x1, x2, · · · , xn . X J f (A) = 0§ K A ? A
λii ÷ v f (λii) = 00.
Nanjing University of Science and Technology 7 / 34Matrix Analysis Dept Math
8. A = (aijij)n×n § ¡ a11 + a22 + · · · + ann A
, § P trA.
9. tr(AB) = tr(BA)).
Nanjing University of Science and Technology 8 / 34Matrix Analysis Dept Math
n×n § e 3 n×n ¦
10. A, B ∈ C P ∈ Cn
P−1AP = B § K ¡ A B q § P A ∼ B § ¡
P q C Ý
"
11. e A ∼ B §