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(3 1 11 3 11 1 3) ?0 00 1? ???-1 0 0 0? ??? ??????? ??????? ??A? D? -1 1? ????? ??? ????? ? ?? ????? ????1/4 3/4
2/3 1/3?
M? det(A-λI) = 0????? det(A-λI) = det?a11-λ a12 a
21a22-λ?
=λ2-(a11+a22)λ+ (a11a22-a12a21) ??det(A-λI) = 0? ??(A-λI)x= 0???? ?? ??????? ??? ??? ??Ax=λ???? ?? ??????? ??? ???x? ?????? ?? ??????? ?????? ??A??????? ? ?? ?????? ??????λ? 2 0? -1-λ3
2-λ=λ(λ+ 1)-6 =λ2+λ-6 = (λ+ 3)(λ-2)?
(A-(-3)I)x=?2 32 3??
x1 x 2? =?0 0? ??2x1+ 3x2= 0
2x1+ 3x2= 0?
-2? -2/3? ?-3 2?
2x2??x2=-23x1? ???? ?? ???????
????x= (x1,x2) = (x1,-2
3x1) =x1(1,-23)?? ???? ?? ???????x= (x1,x2) = (-32x2,x2) =x2(-32,1)?
(A-2I)x=?-3 3
2-2?? x1 x 2? =?0 0? ??-3x1+ 3x2= 0
2x1-2x2= 0?
?1 1? ??????? ? ?? ?? ?????(A-2I)x= 0?x= (x1,x1) =x1(1,1)??x= (x2,x2) =x2(1,1)? ???? ??? ???? (1 0 20 5 03 0 2) det(B-λI) = (5-λ)(λ-4)(λ+ 1)?Sp(B) ={-1,4,5}? ?? ???? ?? ??????? ??????? ???v1= (0,1,0) ??? ?? ??????? ?????? ??????? ?λ= 5?v2= (2,0,3)?? ??????? ?????? ??????? ?λ= 4??v3= (1,0,-1)?? ??????? ?????? ??????? ?λ=-1? y n+1=ayn????? y n=any0? y n+t
100yn=?
1 +t100?
y n??t ?????? ???y4= (1,05)4×1000 = 1215,5?????? ?xn+1=axn+byn y n+1=cxn+dyn????? ?xn+1=qxn+pyn y n+1= (1-q)xn+ (1-p)yn z n+1=?xn+1 y n+1? =?a b c d?? xn y n? =Azn.????? z n=Anz0 ?Av1=λ1v1 Av
2=λ2v2
?[Av1Av2] = [λ1v1λ2v2] ?A[v1v2] = [v1v2]?λ10
0λ2?
?? ??????[v1v2] =P??D=?λ10
0λ2?
AP=PD
A=PDP-1
A n=PDnP-1 v P -1AP=(
10... ...0
0λ20...0
0...0λk-10
0... ...0λk)
)=D? v z n+1=Azn??? z n= (Z0)1? c 1λ n
1v1+ (Z0)2????
c 2λ n
2v2+...+ (Z0)k????
c kλ n kvk ??Z0= ((Z0)1...(Z0)k)t=P-1z0? ?zn= [v1...vk]( n 10
0λn
k) )Z0=c1λn
1v1+c2λn
2v2+...+ckλn
kvk? p(λ) = det(A-λI)? ?? ????A=?-4 2 -1-1? ?pA(λ) =λ2+ 5λ+ 6?Sp(A) ={-3,-2}? ?? ????B=( (4 0-2 0 3 0
3 0-1)
)?pB(λ) = (3-λ)(λ2-3λ+ 2)?Sp(B) ={1,2,3}? ?? ????C=?4 1 -1 2? ?? ????D=?0 2 -1 2? ?pD(λ) =λ2-2λ+ 2?Sp(D) ={1 +i,1-i}? ?? ????E=( (1 3 40 2-1
0 1 2)
)?pE(λ) = (1-λ)(λ2-4λ+ 5)?Sp(E) ={1,2 +i,2-i}? ??λ1+λ2+...+λk=k? i=1λ i=tr(A)? ??λ1×λ2×...×λk=k? i=1λ i= det(A) (4 1 1 11 4 1 11 1 4 11 1 1 4) )? ? ??? ??? ?????? ?????? ??B? ????v= (a,b,c,d)?? ??????? ?????? ??????? ?λ= 3????? (B-3I)v=( (4-3 1 1 1
1 4-3 1 1
1 1 4-3 1
1 1 1 4-3)
(a b c d) (a+b+c+d a+b+c+d a+b+c+d a+b+c+d) (0 0 0 0) (1 0 0 -1) (0 1 0 -1) (0 0 1 -1) -1 2? ??B=?3 00 3? ?????? ?????? ??????λ= 3? (A-3I)v=?1 1 -1-1?? v1 v 2? =?0 0? ?v1 v 2? -1? ? ????B?(B-λI)v=?0 00 0?? v1 v 2? -1 2? -1? v (A-3I)v2=v1??1 1 -1-1?? x y? =?1 -1? ??x y? =?1 0? ?? ?? ?????P= [v1v2] =?1 1 -1 0? P -1AP=?0-1 1 1?? 4 1 -1 2?? 1 1 -1 0? =?3 10 3? ?I? v
2??? ???(A-λ?I)v2=v1? ??P= [v1v2]?????
P -1AP=?λ?1
0λ??
(4 2-4 1 4-3
1 1 0)
(A-2I)v=( (2 2-4 1 2-3
1 1-2)
(x y z) (0 0 0) (1 1 1) (A-3I)v=( (1 2-4 1 1-3
1 1-3)
(x y z) (0 0 0) (2 1 1) v 2?( (1 2-4 1 1-3
1 1-3)
(x y z) (2 1 1) )? ?? ???? ????? ???????v3=( (0 1 0) )????P=( (1 2 01 1 11 1 0) )??P-1AP= (2 0 00 3 10 0 3) (A-λ?I)v2=v1??(A-λ?I)v3=v2? ??P= [v1v2v3]?????P-1AP=( (λ?1 0
0λ?1
0 0λ?)
(3 1 11 2 1 -1-1 1) )? ????? ??????? ?????λ= 2????? ?????? ?????? ??????? (A-2I)v2=v1 (A-2I)v3=v2? ?????? ????P= [v1v2v3]?P-1BP=( (2 1 00 2 10 0 2)
0λ??
Z n+1=?Xn+1 Y n+1? =?λ1
0λ??
Xn Y n?
Xn+1=λXn+Yn??????
Y n+1=λYn?????? X 0=c0 X
1=λX0+c1λ0=λc0+c1
X
2=λX1+c1λ1=λ(λc0+c1) +c1λ=λ2c0+ 2c1λ
X
3=λ3c0+ 3c1λ2
X
4=λ4c0+ 3c1λ2
X n=λnc0+nc1λn-1 X n+1=λ(c0λn+nc1λn-1) +c1λn=c0λn+1+ (n+ 1)c1λn=λXn+Yn. ?Xn Y n? =?c0λn+nc1λn-1 c
1λn?
z n=PZn= [v1v2]?c0rn+nc1rn-1 c 1rn? ?zn= (c0rn+nc1rn-1)v1+c1rnv2.????? ??????? ??????? ???????A=?4 1 -1 2? -1? y n+1=-xn+ 2yn? ?xn=c03n+c1(n3n-1+ 3n) y n=-c0-c1n3n-1? W n+1=( (X n+1 Y n+1 Z n+1) )= (P-1AP)Wn=TWn? ?? ????? ??wn+1=( (x n+1quotesdbs_dbs33.pdfusesText_39