ࠫޅაսඔษંčĎčหᆘᆴ, หᆘཟĎ
၂a؎༯ࢲં൞ڎᆞಒ, ѩඪૼႮğ
(1) ഡ A ൞ n ࢨٚᆔđ A
m
=0đᄵ A ֥หᆘᆴᆺି൞.
(2) ഡ A ൞ n ࢨൌٚᆔđ A
2
= Ađᄵ A ֥หᆘᆴᆺି൞ 1 ࠇ 0.
(3) ഡ X, Y ൞ n ࢨٚᆔ A ֥උႿ҂หᆘᆴ֥หᆘཟđᄵсႵ X
T
Y =0.
(4) ഡ A ൞ n ࢨٚᆔđλ
0
൞ A ֥၂۱หᆘᆴđX
1
,X
2
൞ (λ
0
I − A)X =0֥ࠎԤࢳ༢đᄵ A ֥උ
Ⴟหᆘᆴ λ
0
֥ಆ҆หᆘཟູ k
1
X
1
+ k
2
X
2
đఃᇏ k
1
,k
2
൞۱ၩӈඔ.
(5) ഡ A ൞ 3 ࢨٚᆔđA ֥หᆘᆴູ 0, 0, 1đX
1
,X
2
൞ AX =0֥ࠎԤࢳ༢đX
3
൞ AX = X ֥٤
ࢳđᄵ A ֥ಆ҆หᆘཟູ k
1
X
1
+ k
2
X
2
+ k
3
X
3
đఃᇏ k
1
,k
2
,k
3
൞҂ಆູ֥ၩӈඔ.
(6) ഡ X ൞ n ࢨٚᆔ A ֥หᆘཟđP ൞ n ࢨٚᆔđᄵ P
−1
X ൞ P
−1
AP ֥หᆘཟ.
(7) ഡ X ൞ n ࢨٚᆔ A ֥หᆘཟđ A đᄵ A
−1
X ൞ A
−1
֥หᆘཟ.
(8) ഡ A ൞ n ࢨٚᆔđ A ֥หᆘᆴ൞ 1đᄵ A ა I ཌྷර.
(9) ഡ A ൞ n ࢨٚᆔđ A
2
+ A + I =0đᄵ A ીႵൌ֥หᆘᆴ.
(10) ഡ A ൞ n ࢨٚᆔđ A ཌྷරႿؓ࢘ᆔđᄵ A ֥ n ۱หᆘᆴၳ.
(11) n ࢨᆔ A, B ֥หᆘᆴཌྷđᄵ A ∼ B(ཌྷර).
(12) A ა B ֩ࡎ (ཌྷַ)đᄵ A ∼ B(ཌྷර).
(13) A ∼ B(ཌྷර)đᄵ A ა B ֩ࡎ (ཌྷַ).
(14) A ა B Ⴕ܋֥หᆘᆴ (Їݣᇗඔ) ࠣႵ n ۱ཌྟܱ֥หᆘཟđᄵ
(A) A ∼ B čཌྷරĎĠ(B) A = BĠ(C) |λI − A| = |λI − B|Ġ
(D) |A − B| =0.
č15Ď A, B ֥หᆘᆴٳљູ λ, μđᄵğ
čAĎA
T
ა A Ⴕཌྷ֥หᆘᆴაหᆘཟĠčBĎA + A
T
ࠣ AA
T
֥หᆘᆴٳљູ 2λ ࠣ λ
2
Ġ
čCĎA + B ࠣ AB ֥หᆘᆴٳљູ λ + μ ࠣ λμĠ
čDĎၛഈࢲં҂ᆞಒ.
ؽa༯ᆔڎაؓ࢘ᆔཌྷරđᆔؓ࢘߄่֥ࡱ൞હĤ
1.
⎛
⎜
⎜
⎜
⎝
A
1
A
2
.
.
.
A
s
⎞
⎟
⎟
⎟
⎠
.
2. A ∈ M
n
,r(A)=2,A
2
+ A =0.
aഡ A =(a
ij
)
n×n
đ A đ
n
j=1
a
ij
= a =0(i =1, ··· ,n)đ൫ᆣğa
−1
ູ A
−1
֥၂۱หᆘ
ᆴđѩؓႋ֥၂۱หᆘཟ.
ඹaഡ A, B ∈ M
n
đ A Ⴕ n ۱ၳ֥หᆘᆴ.
ᆣૼğAB = BA ⇐⇒ A ֥หᆘཟ္൞ B ֥หᆘཟ.
aഡ A ູࢨൌؓӫᆔđλ =1, 2, −1 ൞ః۱หᆘᆴđα
1
=(1,a +1, 2)
T
đα
2
=(a −
1, −a, 1)
T
đٳљູ A ֥ؓႋ λ =1, 2 ֥หᆘཟđA
∗
֥หᆘᆴູ λ
0
đ A
∗
β
0
= λ
0
β
0
đఃᇏ β
0
=
(2, −5a, 2a +1)
T
đ a ࠣ λ
0
֥ᆴ.
aഡ A, B ູ n ࢨᆔđᆣૼğAB ა BA Ⴕཌྷ֥หᆘᆴ.
1
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