M n
n −1
P ∈ M n
P
A = (aij ) ∈ M n D · A · P = D A
P
A
P
50 A
P
1 −1 = 3 2 2 1 −1 1 = 4 2 1 1
A50 = (P · D · P −1 ) . . .(50 D
)
−1 1
4 −1 −1
.
.
1
(P · D · P −1 ) = P · D50 · P −1 .
D50 D50
1 = 0
0
0 0
350 0 0 (−2)50
.
A
50
50
−1
= P · D · P
−1 = 4
1 2 1 1
1 −1 1 0 1
0 50
3
1 6 1 2 − 13
− 0 0
50
0
0 (−2)
1 3
− 12
0
1 2
− 13
1
.
A = (aij ) ∈ M n P A (x)
A
P A (x) = |A − xI n |.
1 ( ) = 3 2
P A x
4 −1 2 −1 1 −1
1 − 0 x
0 0 1 0 0 0 1
= (1 −
x)(x2 − x − 6) = (1 − x)(x − 3)(x + 2).
P A (x) = −x3 + 2x2 + 5x − 6. A λ1 P A (x)
A
x
λ1
P A (λ1 ) = 0 A B =
P B (x) = (x − 1)2
1 3
1
−1 0 1
−2
1
B λi
A o(λi )
P A (x)
λi A
1
B
2
λi
A
t
V = (v1 , v2 , . . . , vn )
A
v1 v2
· · vn
=
λi
v1 v2
· · vn
.
λi V
E λ
i
λi
d(λi )
E λ
V
i
E λ
i
d(λi )
= n − rango(A − λi I n).
d(λi )
1
λi
1 ≤ d(λi ) ≤ o(λi ).
V 1 λ1
λ2
A ∈ M n
M n
A
d(λi )
λi
A ∈ M n A
V 2
M n n
= o(λi )
d(1)
= 3−rango(A − d(−2) = 3 − rango(A + 2 · I 3 ) =
A
1 · I 3 ) = 1 = o(1) 1 = o(−2)
d(3)
= 3 − rango(A − 3 · I 3 ) = 1 = o(3)
A d(1)
= 2 − rango(B − 1 · I 2 ) = 2 − rango
0
−1 0 0
=1 = o(1) = 2
B A
λi
A
1 ≤ d(λi ) ≤ o(λi ). 1
λi d(λi )
1 ≤ d(λi ) ≤ o(λi ) = 1
= o(λi ) = 1 n×n
n
P −1 AP = D
P ∈ M n
A D D
P
D A
A D1
1 = 0
D2
−2 = 0
0 0 3 0 0 0 −2 0 0 1 0 0 0 3
.
.
P P λi
(v1 , . . . , vn)
D
t
λi
λi
A
·
· = · ·
v1
vn
λi
·
· · ·
v1
vn
,
(A − λi I n
)·
0 · · = · · · ·
v1
vn
0
.
V =
d(λi )
= o(λi ) 0
1
(v1 , . . . , vn ) o(λi ) P
P A λ1 = 1
0 3
4 −1 1 −1 2 1 −2
0 · = 0 v1 v2 v3
0
(v1 , v2 , v3 ) = (−α, 4α, α) 1
α
,
1 = o(λ1 ) (−1, 4, 1)
1 3
−2 3
4 −1 −1 −1 2 1 −4
0 · = 0 v1 v2 v3
,
0
(v1 , v2 , v3 ) = (β, 2β, β ) β
1
1 = o(3) (1, 2, 1)
3 −2
3 3
4 −1 4 −1 2 1 1
γ
0 · = 0 v1 v2 v3
0
(v1 , v2 , v3 ) = (−γ , γ , γ ) 1
1 = o(−2) (−1, 1, 1)
−2
D1
1 = 0
0 0 3 0 0 0 −2
P 1
−1 = 4
=
,
P 1−1 AP 1 ,
1 −1 2 1 1 1 1
.
D2
−2 0 0 = 0 1 0 = 0 0 3 −1 −1 1 4 2 = 1
P 2−1 AP 2 ,
P 2
1
A
C
1 0 = 0 10 0 = 1
0 −10 0 3 1 1 1 1 = 1 −1
−1 −1 1 0 −1 1 0 1 −1 0 0 1
0 1 0 1 0
0 5 2 0
A
.
1 1
1 0 0 1 0 = 0 −2 −4 −1 1 3 0 = 3 1 0 0 0 −2 1 −1 −1 B
E
1 −1 1 1 −1 −1 A
1
O, X 1 , X 2 , X 3 δ i,j
A = (δ i,j )1≤i,j≤3 OX j OX i l = (l1 , l2 , l3 )
t
t
t = Al P
t = λ n t t n = λ n
n
n
A P A
3 = 1
1 2
1 0 1 2 0
O, X 1 , X 2 , X 3
