✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖
▲➊ ❚❍➚ ❚❍❯ ❍➪❆
❇⑩❖ ❈⑩❖ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
◆●❍■➊◆ ❈Ù❯ ❱➋ ❚➑◆❍ ✣■➋❯ ìẹ
ế
ữợ ❍↔✐ ❚r✉♥❣
✣➔ ◆➤♥❣✱ ✵✶✴✷✵✷✵
▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝ì sð
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺
✶✳✻
✶✳✼
✶✳✽
✶✳✾
✶✳✶✵
✶✳✶✶
▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ✈✉æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ✤ì♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✶ ❙ü ❜➡♥❣ ♥❤❛✉ ❝õ❛ ❤❛✐ ♠❛ tr➟♥
✶✳✻✳✷ P❤➨♣ ❝ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✸ ❚➼❝❤ ♠ët sè ✈ỵ✐ ♠ët ♠❛ tr➟♥ ✳
✶✳✻✳✹ ◆❤➙♥ ❤❛✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✺ ❈❤✉②➸♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✻ ▲ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠❛ tr➟♥ ✳ ✳
✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍↕t ♥❤➙♥ ✈➔ ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
●✐→ trà r✐➯♥❣✱ ✈❡❝t♦ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳
✳
✳
✳
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✹
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✻
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✻
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✼
✽
✽
✾
✶✵
✶✵
✶✶
✶✶
✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔
✶✷
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✸✺
✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷
ỵ ỹ ồ t
ổ t ❧➔ ♠ët ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳
❚➻♠ ❤✐➸✉ ✈➲ ❤➺ ♠ỉ t↔✱ ❝❤♦ t❛ ❝→✐ ♥❤➻♥ tê♥❣ q✉❛♥ ✈➲ ❜↔♥ ❝❤➜t ✈➟♥ ❤➔♥❤ ❝õ❛ ❝→❝
tr↕♥❣ t❤→✐ ♥❣❤✐➺♠ ❝ơ♥❣ ♥❤÷ ♠è✐ ❧✐➯♥ ❤➺ ♠➟t t❤✐➳t ❣✐ú❛ ❝→❝ tr↕♥❣ t❤→✐ tr♦♥❣ t❤í✐
❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳ ❍➺ ♠æ t↔ ✤â♥❣ ✈❛✐ trá ✤✐➲✉ ❦❤✐➸♥ ♠å✐ sü ❜✐➳♥ ✤ê✐ ❝õ❛
♥❣❤✐➺♠ tr♦♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ tæ✐ ✈➔ ❚✐➳♥ s➽ ▲➯ ❍↔✐
❚r✉♥❣ ✤➣ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✧❚➼♥❤ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧
♥❤➡♠ ♠ư❝ ✤➼❝❤ ♥➢♠ rã ❝→❝ q✉② ❧✉➟t ✤✐➲✉ ❦❤✐➸♥ ✈➔ t➻♠ ❤✐➸✉ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ tø♥❣
tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥ ❦❤→❝ ♥❤❛✉✳
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
◆❤➡♠ ❤✐➸✉ rã t❤➜✉ ✤→♦ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ◆❣♦➔✐ r❛ ❝❤ó♥❣
tỉ✐ ♠♦♥❣ ố ữ r ữ t q ợ ử ử ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳
✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✳
❚➼♥❤ ❈✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ■✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔✱
♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❜❛ tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥ tr➯♥ ❝õ❛ ❤➺ ♠æ t↔ tr♦♥❣ Rn.
✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ❤➺ ♠æ t↔ ❜✐➳♥ t❤ü❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn✱ ❜❛ tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥
C − R − I ❝õ❛ ❤➺ ♠ỉ t↔ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳
✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
❈❤ó♥❣ tỉ✐ sû ❞ư♥❣ ữỡ ự ỵ tt tr q tr tỹ
t rữợ t ú tổ t t ồ ừ ỳ t
trữợ q ✤➳♥ t➼♥❤ ✤✐➲✉ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ❚ø ✤â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣
t÷ì♥❣ tü ❤â❛✱ ❝❤ó♥❣ tỉ✐ ❦❤→✐ q✉→t ♥❤ú♥❣ ❦➳t q✉↔ ✤â✱ ❝❤ó♥❣ tỉ✐ s➩ ✤÷❛ r❛ ♥❤ú♥❣
❦➳t ❧✉➟♥ ♠ỵ✐ ❝❤♦ ✤➲ t➔✐✳
✸
❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝ì sð
✶✳✶ ▼❛ tr➟♥
▼❛ tr➟♥ tr➯♥ tr÷í♥❣ K ❧➔ ♠ët ♠↔♥❣ ❝❤ú ♥❤➟t ❧✐➺t ❦➯ ❝→❝ sè ♥❤÷✿ sè t❤ü❝✱ sè ♣❤ù❝
❤❛② ♠ët ❤➔♠ sè✱✳✳ ✤÷đ❝ ①➳♣ t ởt trt t t ỗ m n ❝ët✳ ❑➼
❤✐➺✉✿
a11
a12
... a1n
a21
A=
...
a22
... a2n
am1 am2 ... amn
m.n
❚r♦♥❣ ✤â✿
aij ✿ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ♠❛ tr➟♥ ð ❞á♥❣ t❤ù i ✈➔ ❝ët j
i✿❝❤➾ sè ❞á♥❣✳
j ✿❝❤➾ số ởt
m.n ồ tữợ tr tữớ ❝→❝ ❝❤ú ❝→✐ A, B, C... ✤➸ ❦➼ ❤✐➺✉ ❝→❝ ♠❛
tr➟♥✳
❉↕♥❣ t❤✉ ❣å♥ A = [aij ]mn
❱➼ ❞ö ✶✳✶✳ ❈❤♦ ♠❛ tr➟♥
A=
9 −13
1
20 5
❧➔ ♠❛ tr➟♥ ❝ï 2.3 ✭✷ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳
✹
−6
✶✳✷ ▼❛ tr➟♥ ✈✉æ♥❣
▼❛ tr➟♥ ✈✉æ♥❣ ❧➔ ♠❛ tr➟♥ ❝â sè ❤➔♥❣ ✈➔ sè ❝ët ❜➡♥❣ ♥❤❛✉✳ ▼❛ tr➟♥ ❝➜♣ n.n ✤÷đ❝
❣å✐ ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n✳ ❈→❝ ♣❤➛♥ tû aij , (i = j) t↕♦ t❤➔♥❤ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤
❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣✳
❱➼ ❞ư ✶✳✷✳ ▼❛ tr➟♥
1 1 −1
B = 2 3 −2
0 2
1
❧➔ ♠❛ tr➟♥ ✈✉è♥❣ ❝➜♣ 3 ✭✸ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳ ❈→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❧➛♥
❧÷đt ❧➔ 1; 3; 1.
✶✳✸ ▼❛ tr➟♥ ✤ì♥ ✈à
▼❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n tr♦♥❣ ✈➔♥❤ ❱ ❧➔ ♠➔ tr➟♥ ✈✉æ♥❣ ❝➜♣ n tr♦♥❣ ✤â t➜t ❝↔ ❝→❝
♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ ✶✱ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ❜➡♥❣ ❦❤æ♥❣✳
1 0 . . 0
0 1 . . 0
En =
. . . . .
0 0 . . 1
✶✳✹ ▼❛ tr➟♥ ❝♦♥
▼❛ tr➟♥ ❝♦♥ ❝õ❛ A ❧➔ ♠❛ tr➟♥ B ✤÷đ❝ t❤➔♥❤ ❧➟♣ tø ♠❛ tr➟♥ ❜❛♥ ✤➛✉ ❜➡♥❣ ❝→❝❤ ❜ä
✤✐ ♠ët sè ❞á♥❣✱ ✈➔ ♠ët sè ❝ët✳
1 1
❱➼ ❞ö ✶✳✸✳ ❈❤♦ ♠❛ tr➟♥ A = 2
❝♦♥ ❝õ❛ ♠❛ tr➟♥ A.
2
0 −1 .
❚❛ ❝â ♠❛ tr➟♥ B = 2
3 1 −1
✶✳✺ ▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤
❈❤♦
1 1
0
❧➔ ♠❛ tr➟♥
3 1
❧➔ ♠❛ tr➟♥ ✈✉è♥❣ tr➯♥ tr÷í♥❣ K✱ ♥➳✉ tỗ t ởt tr B s
A ì B = En ✳ ❑❤✐ ✤â A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✱ B ❧➔ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦
A
✺
❝õ❛ ♠❛ tr➟♥ A. ✈➼ ❞ư ✈➔ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦
✶✳✻ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥
✶✳✻✳✶ ❙ü ❜➡♥❣ ♥❤❛✉ ❝õ❛ ❤❛✐ ♠❛ tr➟♥
❍❛✐ ♠❛ tr➟♥ ❜➡♥❣ ♥❤❛✉ ❦❤✐ ❝â ❝→❝ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❜➡♥❣ ♥❤❛✉ tứ ổ ởt
õ ũ tữợ
A = (aij )m×n ✈➔ B = (bhk )p×q
m=p
❑❤✐ ✤â A = B ⇔
n=q
aij = bhk
✶✳✻✳✷ P❤➨♣ ❝ë♥❣
✣à♥❤ ỵ tr A B õ ũ tữợ n ì m ờ A + B
ởt tr ũ tữợ tr õ tỷ tr♦♥❣ ✈à tr➼ t÷ì♥❣ ù♥❣ ❜➡♥❣ tê♥❣
❤❛✐ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❝õ❛ ♠é✐ ♠❛ tr➟♥✿
(A + B)ij = Aij + Bij , 1 ≤ i ≤ m,
✈➔1 ≤ i ≤ n
❱➼ ❞ö ✶✳✹✳
1 3 1
1 0 0
+
0 0 5
7 5 0
=
1+0 3+0 1+5
1+7 0+5 0+0
=
1 3 6
8 5 0
❚➼♥❤ ❝❤➜t
✶✳ A + B = B + A
✷✳ (A + B) + C = A + (B + C)
✸✳ 0 + A = A + 0 = A
✹✳ A + (−A) = (−A) + A = 0
✶✳✻✳✸ ❚➼❝❤ ♠ët sè ✈ỵ✐ ởt tr
ỵ cA ừ ởt số c ✈ỵ✐ ♠❛ tr➟♥ A ❧➔ ♠ët ♠❛ tr➟♥ ❝â ữủ
ộ tỷ ừ A ợ c
(cA)ij = c.Aij
✻
❱➼ ❞ö ✶✳✺✳
2.
1
−3
8
4 −2
5
=
2.1
2.8
2(−3)
2.4 2(−2)
2.5
=
2 16 −6
8 −4 10
❚➼♥❤ ❝❤➜t
✶✳ a.(A + B) = a.A + a.B
✷✳ (a + b).A = a.A + b.A
✸✳ a.(b.A) = (ab).A
✹✳ 1.A = A
✶✳✻✳✹ tr
ỵ tr A = (aij )m×n ❝â ❝➜♣ m × n ✈➔ ♠❛ tr➟♥ B = (bij )n×p ❝â
❝➜♣ n × p✳ ❚➼❝❤ ❝õ❛ ❤❛✐ ♠❛ tr➟♥ A ✈➔ B ❧➔ ♠ët ♠❛ tr C = (cij )mìp m ì p ợ
n
ckh =
aki bih ,
j=1
✈ỵ✐ k = 1, m, h = 1, p.
ữủ tr A ợ tr B✿ ❙è ♣❤➛♥ tû tr➯♥ ❞á♥❣ ❝õ❛
♠❛ tr➟♥ A ♣❤↔✐ ❜➥♥❣ sè ♣❤➛♥ tû tr➯♥ ❝ët ❝õ❛ ♠❛ tr➟♥ B t÷ì♥❣ ù♥❣✳
❱➼ ❞ư ✶✳✻✳ ❈❤♦ A =
1 0
1
0 1 −1
✈➔ B = 0
1.1 + 0.0 + 1.1
0
1
1 −1
❑❤✐ ✤â
C = A.B =
1
1.0 + 0.1 + 1(−1)
0.1 + 1.0 + (−1).1 0.0 + 1.1 + (−1)(−1)
❚➼♥❤ ❝❤➜t
✶✳ (A + B).C = A.C + B.C
✷✳ A(B + C) = A.B + A.C
✸✳ (A.B).C = A.(B.C)
✹✳ E.A = A.E = A ✭✈ỵ✐ E ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à✮
✺✳ (AB)T = B T AT ✳
✻✳ P❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ❦❤æ♥❣ ❝â t➼♥❤ ❣✐❛♦ ❤♦→♥✳
✼
=
2
−1
−1
2
❱➼ ❞ö ✶✳✼✳ ❈❤♦ A =
1 2
3 4
✈➔ B =
AB =
✈➔
BA =
0 −1
6
7
1 2
0 −1
3 4
6
7
0 −1
1 2
6
3 4
7
=
=
12 13
24 25
1 4
27
40
AB = BA.
ỵ ởt tr õ tữợ m ì n ✈ỵ✐ ❝→❝ ❣✐→ trà aij t↕✐
❤➔♥❣ i✱ ❝ët j t❤➻ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à B = AT ❧➔ ♠❛ tr õ tữợ n ì m ợ
tr bij = aji✳
❱➼ ❞ö ✶✳✽✳
T
1
2
3
1
0
= 2 −6
0 −6 7
3
7
❚➼♥❤ ❝❤➜t
✶✳ (A + B)T = AT + B T
✷✳ (a.A)T = a.AT
✸✳ (AB)T = B T AT
✹✳ (AT )T = A
✶✳✻✳✻ ▲ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠❛ tr➟♥
❈❤♦ A ❧❛ ♠ët tr ổ tr trữớ K A ì A ữủ ỵ A2
ữỡ ừ tr A
ợ n số ữỡ tũ ỵ t t A ì A ì ... ì A ỵ An ữủ ồ
ụ tứ n ừ A
ữợ A0 = E
✽
0 1 0
❱➼ ❞ö ✶✳✾✳ ❈❤♦ A = 0
0 1
0 0 0
0 0 1
⇒ A2 = 0 0 0
0 0 0
0 0 0
✈➔ A3 = 0
0 0
0 0 0
✶✳✼ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥
✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❧➔ tê♥❣ ✤↕✐ sè ❝õ❛ n! sè ❤↕♥❣✱ ♠é✐ sè ❤↕♥❣ ❧➔
t➼❝❤ ❝õ❛ n ♣❤➛♥ tû ❧➜② tr➯♥ ❝→❝ ❤➔♥❣ ✈➔ ❝→❝ ❝ët ❦❤→❝ ♥❤❛✉ ❝õ❛ ♠❛ tr➟♥ A✱ ộ t
ữủ ợ tỷ +1 −1 t❤❡♦ ♣❤➨♣ t❤➳ t↕♦ ❜ð✐ ❝→❝ ❝❤➾ sè ❤➔♥❣ ✈➔
❝❤➾ sè ❝ët ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ t➼❝❤✳ ●å✐ Sn ❧➔ ♥❤â♠ ❝→❝ ❤♦→♥ ✈à ❝õ❛ ❝õ❛ n ♣❤➛♥
tû 1, 2, ..., n✱ t❛ ❝â ✭❝æ♥❣ t❤ù❝ ▲❡✐❜♥✐③✮
n
ai,δ(i)
sng(δ)πi=1
det(A) =
δ∈Sn
⑩♣ ❞ư♥❣ ❝❤♦ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ ✶✱✷✱✸ t❛ ❝â✿
det[a] = a
det
a11 a12
a21 a22
a11 a12 a13
= a11 a22 − a12 a21
det a21 a22 a23 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32
a31 a32 a33
− a13 a22 a31 − a12 a21 a33 − a11 a23 a32
−2 2 −3
❱➼ ❞ö ✶✳✶✵✳ ❚➼♥❤ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A = −1
2
1
3
0 −1
det (A) = (−2).1(−1) + 2.3.2 + (−3).(−1).0 − 2.1.(−3) − 0.3.(−2) − (−1).(−1).2
= −2 + 12 + 0 + 6 + 0 − 2
= 18
✾
❚➼♥❤ ❝❤➜t
✶✳ det (AB) = det (A) det (B) = det (B) det (A).
✷✳ ▼❛ tr➟♥ A ❦❤↔ ♥❣❤à❝❤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤à♥❤ t❤ù❝ ❝õ❛ A ❦❤→❝ 0✱ t❛ ❝â✿
det (A−1 ) = det (A)−1 .
✸✳ det (AT ) = det (A).✳
✶✳✽ ▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤
▼❛ tr➟♥ A ✈✉æ♥❣ ❝➜♣ n ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ♥❣❤à❝❤ ✭❦❤ỉ♥❣ s✉② ❜✐➳♥✮ tr V
tỗ t tr A ũ s❛♦ ❝❤♦ AA = A A = E ✳ ❑❤✐ ✤â ❆✬ ✤÷đ❝ ❣å✐ ❧➔ ♠❛
tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ❆✱ A1
ữợ t tr
✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A✿
✲ ◆➳✉ det(A) = 0 t❤➻ A ❦❤æ♥❣ ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳
✲ ◆➳✉ det(A) = 0 t❤➻ A ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳
✷✳ ▲➟♣ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à AT ❝õ❛ A✳
✸✳ ▲➟♣ ♠❛ tr➟♥ ♣❤ư ❤đ♣ AT : A∗ = (ATij )nn✳
1
A∗ .
✹✳ ❚➼♥❤ ♠❛ tr➟♥ A−1 = det(A)
❱➼ ❞ö ✶✳✶✶✳ ❈❤♦ A =
✶✳ det(A) =
✷✳ AT =
✸✳ A∗ =
1 −2
3
1
2
1 −2
3
2
✳ ❚➼♥❤ A−1
=8
3
−2 2
−4 2
−6 1
✹✳ A−1 = 81
2
2
−3 1
=
−0, 5
0, 25
−0, 75 0, 125
✶✳✾ ❍↕t ♥❤➙♥ ✈➔ ↔♥❤
◆➳✉ V ✈➔ W ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡✈t♦ tr➯♥ ❝ị♥❣ ♠ët tr÷í♥❣✱ t❛ ♥â✐ r➡♥❣ →♥❤ ①↕
f : V → W ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â ❤↕t ♥❤➙♥ ừ f ỵ ker f
ừ f ỵ im(f ) ữủ ♥❣❤➽❛ ♥❤÷ s❛✉✿
ker f = {x ∈ V : f (x) = 0}
imf = {f (x) : x ∈ V}
❚➼♥❤ ❝❤➜t
dim(ker(f )) + dim(im(f )) = dim(V )
✶✳✶✵ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥
❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ A ❧➔ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ ❝→❝ ✤à♥❤ t❤ù❝ ❝♦♥ ❦❤→❝ ✵ ❝â tr♦♥❣ A✳ ỵ
rank(A) r(A)
1 2 3 4
ử ❈❤♦ ♠❛ tr➟♥ A = 2
4 6 8
3 5 7 9
▼❛ tr➟♥ ❝♦♥ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ A ❧➔
2 3 4
A234
123 = 4 6 8
5 7 9
♥➯♥ rank(A) = 3.
✶✳✶✶ ●✐→ trà r✐➯♥❣✱ ✈❡❝t♦ r✐➯♥❣
❈❤♦ A ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n tr➯♥ tr÷í♥❣ sè K (K = R, C)✳ ❙è λ ∈ K ✤÷đ❝ ❣å✐ ❧➔
❣✐→ trà r✐➯♥❣ ✭trà r ừ tr A tỗ t ởt t u = 0, u ∈ Kn s❛♦ ❝❤♦
Au = λu
❑❤✐ ✤â ✈❡❝t♦ u ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝t♦ r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ
✶✶
❈❤÷ì♥❣ ✷
❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛
❤➺ ♠ỉ t↔
✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉
❚✐➳♥ ❤➔♥❤ ①❡♠ ①➨t ❤➺ ♠æ t↔ ❝â ❞↕♥❣✿
F (t, x, x,
˙ u) = 0,
x(t0 ) = x0 ,
✭✷✳✶✮
y − G(x, t) = 0.
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳
✶✳ ❍➺ ♠ỉ t↔ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❤♦➔♥ t♦➔♥ ✭✤✐➲✉ ❦❤✐➸♥ ❈✮ ♥➳✉ ✈ỵ✐
♠å✐ tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ ❝❤♦ trữợ x0 Rn tr t t tú xf Rn
tỗ t ởt ❜✐➳♥ ✤ê✐ ❤➺ tø tr↕♥❣ t❤→✐ x0 ✤➳♥
tr↕♥❣ t❤→✐ xf tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ tf ≥ t0 ✭∃u, tf < ∞ s❛♦ ❝❤♦
x(tf , u, x0 ) = xf ✮✳
✷✳ ❚r♦♥❣ ❤➺ ♠æ t↔ ✭✷✳✶✮✱ t❛ ❦➼ ❤✐➺✉ Rx t õ t t tợ ữủ tứ x0 ∈ Rn
♥➳✉ ✈ỵ✐ t➜t ❝↔ ♠å✐ xf ∈ Rx ✱ tỗ t ởt ữủ ởt
❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✤➸ ❞à❝❤ ❝❤✉②➸♥ ❤➺ tø tr↕♥❣ t❤→✐ x0 ✤➳♥ tr↕♥❣ t❤→✐
xf tr♦♥❣ t❤í✐ ❣✐❛♥ ❣✐ỵ✐ ❤↕♥ ✭∃u, tf < ∞ s❛♦ ❝❤♦ x(tf , u, x0 ) = xf ∈ Rx ✮✿
0
0
0
Rx0 := {xf ∈ Rn | ∃u, tf < ∞ : x(tf , u, x0 ) = xf } ⊆ Rn .
✶✷
✣➦t R := x ∈χ Rx ✈ỵ✐ χtc ⊆ Rn ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❣✐→ trà ✤➛✉ x0 t↕✐
t❤í✐ ✤✐➸♠ t0✳ ❍➺ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ tr♦♥❣ t➟♣ ❝â t❤➸
✤↕t ✤÷đ❝ ✭❘✲✤✐➲✉ ❦❤✐➸♥✮✱ ♥➳✉ ♠å✐ tr↕♥❣ t❤→✐ tr♦♥❣ R ✤➲✉ ❝â t❤➸ ✤↕t ✤÷đ❝
tø ♥❤✐➲✉ tr t trữợ x0 tr tớ ỳ ❤↕♥ ✭❝❤♦
x0 ∈ R, xf ∈ R, ∃u, tf < ∞ s❛♦ ❝❤♦ x(tf ; u, x0 ) = tf
0
t0
c
0
0
ú ỵ
õ t ữủ ổ ❦❤✐ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✤ë♥❣ ❧ü❝ ✤✐➲✉
❦❤✐➸♥ ❤ú✉ ❤↕♥✳
✷✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ❤➺ ❝â t❤➸ ❧➔ ổ ữủ
t ợ ỳ r ở sè ❝è ✤à♥❤ ♥❣❤✐➺♠ tr➯♥ ♠ët ✤❛ t↕♣ ✤➣ ❜✐➳t✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ E = In✱ ❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ữủ trũ ợ
t
ử t ❤➺ ♠æ t↔ s❛✉
0 0
x˙ 1
1 0
x˙ 2
=
0 1
x1
1 0
x2
+
0
1
u
❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝❤♦ ❜ð✐ R = {(x1 , x2 ) ∈ R2 |x2 = 0} ✈➔ ❤➺ tr➯♥ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥
❝â t❤➸ ✤↕t ✤÷đ❝✳
❚r♦♥❣ ❤➺ ♠ỉ t↔ ❤✐➺♥ t÷đ♥❣ ❦❤→❝ ♣❤→t s✐♥❤ ♥➳✉ ❤➔♠ ✤➛✉ ✈➔♦ ❧✐➯♥ tö❝ ❞✉② ♥❤➜t✳ ❱➻
♥❣❤✐➺♠ ❝â t❤➸ ♣❤ö t❤✉ë❝ ✈➔♦ ✤↕♦ ❤➔♠ ❝õ❛ ✤➛✉ ✈➔♦ ♥➯♥ ❝â t❤➸ ①↔② r❛ ✈✐➺❝ ổ
õ ờ tỗ t ♠ỉ t↔ ❝â t❤➸ ♥❤➟♥ ♠ët ♥❣❤✐➺♠ s✉② rë♥❣✳
❱➼ ❞ư ✷✳✷✳ ❳➨t ❤➺ ♠æ t↔ s❛✉
0 1
x˙ 1
0 0
x˙ 2
=
0 1
x1
1 0
x2
+
1
1
ợ u ữủ
u(t) =
0, 0 ≤ t ≤ 1,
1, 1 ≤ t ≤ tf
u ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤❡♦ t✳ ◆❣❤✐➺♠ ❝õ❛ ❤➺ ✤➣ ❝❤♦ ✤÷đ❝ ❝❤♦ ❜ð✐✿
x1 (t) = u + u˙
x2 (t) = u.
✶✸
u
trữợ ổ õ ờ tỗ t tr t
ỗ ừ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✈➔ (x1, x2) ❧➔ ♠ët ♥❣❤✐➺♠ t❤❡♦
♥❣❤➽❛ s✉② rë♥❣✳
❍➻♥❤ ✷✳✶✿ ❚r↕♥❣ t❤→✐ ①✉♥❣ ❝õ❛ ♥❣❤✐➺♠✳
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ▼ët ❤➺ ♠æ t↔ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ①✉♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✭✤✐➲✉
❦❤✐➸♥ ■✮ ♥➳✉ ợ ồ tr t trữợ x0 Rn tỗ t↕✐ ♠ët ✤➛✉ ✈➔♦ ✤✐➲✉ ❦❤✐➸♥
✤÷đ❝✱ ✤✐➲✉ ✤â ❧➔♠ ❜✐➳♥ ✤ê✐ ❤➺ t❤➔♥❤ ❝→❝ tr↕♥❣ t❤→✐ ①✉♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳
◆â ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ t➼♥❤ ✤✐➸✉ ữủ ừ ỹ tữỡ ữỡ
ợ õ ❜ä t➜t ❝↔ ❝→❝ tr↕♥❣ t❤→✐ ①✉♥❣ ❧ü❝ ❜➡♥❣ ❝→❝❤ ❝❤å♥ ✉ t❤➼❝❤ ❤đ♣✳
❱✐➺❝ ♥➔② ❝â t❤➸ ✤÷đ❝ t❤ü❝ ❤✐➺♥ tr t ữủ ợ ộ tr
t trữợ x0 tỗ t ởt tr t ♥❣÷đ❝ s❛♦ ❝❤♦ ❤➺ ❧➦♣ ❦❤➨♣ ❦➼♥
❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ①✉♥❣ ❧ü❝ ♥➔♦✮✳
❈❤♦ ❤➺ ♠ỉ t↔ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❝❤➼♥❤ q✉② ❝â ❞↕♥❣
E x˙ = Ax + Bu,
x(0) = x0
y = Cx
✭✷✳✷✮
✈ỵ✐ E, A ∈ Rn,m, B ∈ Rn,m, C Rp,n tỗ t sỹ ổ t trữ sè ✤ì♥ t❤✉➛♥
❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤→❝ ♥❤❛✉✳ ❑❤ỉ♥❣ ♠➜t ✤✐ t➼♥❤ tê♥❣ q✉→t✱ ❝❤ó♥❣
t❛ ❣✐↔ sû r r = rank(E) < n õ tỗ t ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥ T, W ∈ Rn,n
s❛♦ ❝❤♦ tữỡ ữỡ ợ
x 1 = Jx1 + B1 u,
x1 (0) = x1,0
N x˙ 2 = x2 + B2 u,
x2 (0) = x2,0
y = C 1 x1 + C 2 x2 .
✶✹
✭✷✳✸❛✮
✭✷✳✸❜✮
✭✷✳✸❝✮
✈ỵ✐
W ET =
CT =
Inf
0
0
N
, W BT =
C1 C2
J
, W AT =
B1
B2
0
0 In∞
, T −1 x =
,
x1
x2
✈➔ ❝❤♦ v = ind(E, A)✳ ❚❛ ❣å✐ ✭✷✳✸❛✮ ❧➔ ❤➺ ❝♦♥ ❝❤➟♠ ❝õ❛ nf ✈➔ ✭✷✳✸❜✮ ❧➔ ❤➺ ❝♦♥ ♥❤❛♥❤
❝õ❛ n∞✳
❑❤✐ ✤â✱ t❛ ❜✐➳t tr t ỗ ừ
t
x1 (t) = eJt x1 (0) +
eJ(t−s) B1 u(s)ds
(t > 0)
0
v−1
N i B2 u(i) .
x2 (t) = −
i=0
❱➻ ✈➟②✱ ♠ët ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✭❝❤♦ ♠ët ♥❣❤✐➺♠ ❝ê ✤✐➸♥✮ ♣❤↔✐ t❤ä❛ ♠➣♥ u ∈
T
Cp v−1 (I, Rm )✳ ❱ỵ✐ ♠å✐ t > 0✱ x(t) = T x1 T x2 T
✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t
x1(0) trữợ u(s) 0 ≤ s ≤ t✱ ✈➔ t❤í✐ ✤✐➸♠ t✳ ❚r♦♥❣
tr÷í♥❣ ủ t trữợ x2(0) ữủ ♥❤➜t q✉→♥ ✭x2(0) ✤÷đ❝
①→❝ ✤à♥❤ ❞✉② ♥❤➜t✮ ✈➔ ❝❤➾ ❞✉② t x1(0) ợ õ t ữủ ồ ởt tũ þ✳
❈❤ó♥❣ t❛ ❦➼ ❤✐➺✉ R˜0 ❧➔ t➟♣ ❝â t❤➸ ✤↕t ữủ ừ tứ
trữợ x1(0) = 0 ✭✈➔ x2(0) ♥❤➜t q✉→♥✮✳
❇ê ✤➲ ✷✳✶✳ ❱ỵ✐ ♠å✐ ✤❛ t❤ù❝ p(t) = 0✱ ①➨t ♠❛ tr➟♥
t
T
p(s)eA1 s B1 B1 T eA1 s p(s)ds,
W (p, t) =
0
✈ỵ✐ A1 ∈ Rn,n, B1 ∈ Rn,m✳ ❑❤✐ ✤â✿
Im(W (p, t)) = Im
A1 A1 B1 ... A1 n−1 B1
✈ỵ✐ ♠å✐ t > 0
❈❤ù♥❣
ờ tr tữỡ ữỡ ợ
n1
ker(B1T (AT1 )i )
ker(W (p, t)) =
i=0
✶✺
❈❤♦ x ∈ ker(W (p, t))✱ ❦❤✐ ✤â
t
xT W (p, t)x =
0
✈➻ ✈➟② B1T eA
t❛ ❝â
T
1s
t
T
T
B1T eA1 s p(s)x
xT p(s)eA1 s B1 B1T eA1 s p(s)xds =
0
p(s)x = 0
2
ds = 0,
2
✈ỵ✐ 0 ≤ s ≤ t✳ ✣❛ t❤ù❝ p(s) ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ tr♦♥❣ [0, t]✱
T
B1T eA1 s x = 0, 0 ≤ s ≤ t.
T
T i ♥➯♥ ker(W (p, t)) ⊆ ker(B T (AT )i ✳ ❱ỵ✐
❱➻ s tị② þ✱ ♥➯♥ t❛ ❝â x ∈ n−1
1
1
i=0 ker B1 (A1 )
T
T
i
x ∈ ker(B1 (A1 ) )✱ t✐➳♥ ❤➔♥❤ ♥❣÷đ❝ ❧↕✐ q✉→ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝
x ∈ ker(W (p, t))✱ ❝â ♥❣❤➽❛ ❧➔
ker(B1T (AT1 )i ) ⊆ ker(W (p, t))
❱➟②
ker(B1T (AT1 )i ) = ker(W (p, t))
◆❣❤➽❛ ❧➔
Im(W (p, t)) = Im
A1 A1 B1 ... A1 n−1 B1
❇ê ✤➲ ✷✳✷✳ ❈❤♦ xi ∈ Rn, i = 0, 1, ..., v − 1 t1 > 0 õ tỗ t ởt ✤❛ t❤ù❝
p(t) ∈ Rn
❜➟❝ v − 1 s❛♦ ❝❤♦ p(i)(t) = xi ✈ỵ✐ i = 0, 1, ..., v − 1
❈❤ù♥❣ ♠✐♥❤✳
❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ❝→❝❤ ✤➦t
p(t) = x0 + x1 (t − t1 ) +
1
1
x2 (t − t1 )2 ... +
xv−1 (t − t1 )v−2
2!
(v − 1)!
⇒ p(t1 ) = x0 , p (t1 ) = x1 , p (t2 ) = x2 , ...
ỵ R˜0 ❧➔ t➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝õ❛ ✭✷✳✸✮ tø
trữợ x1(0) = 0 õ
R0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] .
ú ỵ r ỵ õ tữỡ tỹ ữ t srts = ì
❈❤♦ x1(0) = 0, t > 0✱ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✮ ✤÷đ❝ ❝❤♦ ❜ð✐
❈❤ù♥❣ ♠✐♥❤✳
v−1
t
J(t−s)
x1 (t) =
e
N i B2 u(i) (t).
x2 (t) = −
B1 u(s)ds,
0
i=0
❍✐➸♥ ♥❤✐➯♥✱ x2(t) ∈ Im[ B2 N B2 ... N n −1B2] ✭✈➻ v ≤ n∞✮✳ ❍ì♥ ỳ tỗ t
t i(t) R, i = 0, ..., nf − 1 s❛♦ ❝❤♦✿
∞
eJt (t) = β0 (t)I + β1 (t)I + ... + βnf −1 (t)J nf −1
❱➟②✱
nf −1
t
J(t−s)
x1 (t) =
e
t
βi (t − s)u(s)ds ∈ Im[B1 JB1 ...J nf −1 B1 ]
J i B1
B1 u(s)ds =
0
0
i=0
✈ỵ✐ ♠å✐ t > 0✳ ❱➟②✱
x(t) =
x1 (t)
∈ Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ]
x2 (t)
▼➦t ❦❤→❝✱ ✤➦t
xˆ =
xˆ1
xˆ2
∈ Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] ,
✈ỵ✐ xˆ1 ∈ Im[ B1 JB1 ... J n 1B1] x2
tỗ t wi Rn , i = 0, 1, ..., v − 1 s❛♦ ❝❤♦
f
∈ Im[ B2 N B2 ... N n∞ −1 B2 ] .
◆➯♥
∞
v−1
N i B2 wi .
xˆ2 = −
i=1
❚ø ❜ê ✤➲ ✭✷✳✷✮✱ ✈ỵ✐ ồ t > 0 ố tỗ t ởt t❤ù❝ p(s) ❜➟❝ v − 1 s❛♦ ❝❤♦
p(i) (t) = wi . ◆➯♥✱ sû ❞ö♥❣ ❤➔♠ ✤➛✉ ✈➔♦ u(t) = u1 (t) + p(t) t❛ ✤÷đ❝ ❤➺
t
t
J(t−s)
x1 (t) =
e
0
✈➔
eJ(t−s) B1 p(s)ds
B1 u(s)ds +
0
t
eJ(t−s) B1 p(s)ds ∈ Im[ B1 JB1 ... J nf −1 B1 ]
x˜1 := xˆ1 −
0
✈ỵ✐ t > 0 ❝è ✤à♥❤✳ ❱ỵ✐ ♠å✐ t > 0 ❝è ✤à♥❤✱ ✤➦t q(s) = sv (s − t)v = 0✳ ❚ø ờ t
s r sỹ tỗ t ừ z ∈ Rn s❛♦ ❝❤♦ W (q, t)z = x˜1✳ ✣➦t u1(s) = q(s)2B1T eJ (t−s)z
T
f
✶✼
ợ 0 s t t ữủ ỗ
t
J(ts)
x1 (t) =
e
B1 q(s)
2
t
T
B1T eJ (ts) zds +
eJ(ts) B1 p(s)ds
0
0
t
J(ts)
=
q(s)e
T
B1 B1T eJ (t−s) q(s)dsz
t
eJ(t−s) B1 p(s)ds
+
0
0
= W (q, t) + xˆ1 − x˜1 = xˆ1
✈➔
v−1
v−1
i
(i)
N i B2 (u1 (t) + p(i) (t))
N B2 u (t) = −
x2 (t) = −
i=0
i=0
v−1
N i B2 wi = xˆ2
=−
i=0
❱➟②✱ xˆ ∈ R˜0.
❱➼ ❞ö ✷✳✸✳
✶✳ ❳➨t ❤➺ s❛✉
x˙1 =
1 1
0 1
0 = x2 +
0
x1 +
1
−1 0
x1 (0) = x01 ,
u,
u,
✈ỵ✐ n = 4, nf = 2, n∞ = 2✱ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ð ❞↕♥❣ ❝❤➼♥❤ t➢❝✳ ❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ tø
x1 (0) = 0 ✤÷đ❝ ❝❤♦ ❜ð✐
R˜0 = Im[ B1 JB1 ] ⊕ [ B2 N B2 ] = R2 ⊕ (R ⊕ {0}) = R3 ⊕ {0}.
✷✳ ❳➨t ❤➺ s❛✉
0 1
1 0
−1
x˙2 = x2 +
−1
❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝❤♦ ❜ð✐ R˜ = R2 tr t ỗ ừ ữủ ❝❤♦
❜ð✐
v−1
N i B2 u(i) (t) =
x2 (t) = −
i=0
u + u˙
u
❱➟② ♥➯♥✱ ✈ỵ✐ ♠å✐ w = [ w1 w2 ]T ∈ R2 ✈➔ t > 0 t❛ ❝â t❤➸ ❝❤å♥ tø u(t) s❛♦ ❝❤♦
u(t1 ) = w2 , u(t
˙ 1 ) = w1 − w2 ✈➔ t❛ ✤÷đ❝
x2 (t) =
✶✽
w1
w1
.
ữ ỵ r ởt ữỡ tự õ t ỏ ọ ởt ữủ
ỡ ợ ||w1 − w2|| ❝➔♥❣ ❧ỵ♥ t❤➻ u˙ ❝➔♥❣ ❧ỵ♥✳ ✣➸ sû ử t q t t
t ỵ t❤✉②➳t ✤✐➲✉ ❦❤✐➸♥ s❛✉✳
✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔
❇ê ✤➲ ✷✳✸✳ ✭❇ê ✤➲ ❍❛✉t✉s✲P♦♣♦✈✮ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
✶✳ ❍➺ x˙ = Ax + Bu ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
✷✳ rank(K) = rank[ B AB ... An−1B ] = n
✸✳ ◆➳✉ z ❧➔ ✈❡❝t♦ r✐➯♥❣ ❝õ❛ AT ✱ t❤➻ zT B ∈ C
✹✳ rank[ λI − A B ] = n ✈ỵ✐ ♠å✐ λ ∈ C
✣à♥❤ ỵ
t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉
rank[ λI − A B ] = n, ∀λ ∈ C, λ
❤ú✉ ❤↕♥
✷✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ ❝♦♥ ♥❤❛♥❤ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
❜✮ rank(K) = rank[ B N B ... N n−1B ] = n∞
❝✮ rank[ N B2 ] = n∞
❞✮ rank[ E B ] = n
❡✮ ❱ỵ✐ ♠å✐ ♠❛ tr➟♥ ❦❤ỉ♥❣ s✉② ❜✐➳♥ Q1 ✈➔ P1 t❤ã❛ ♠➣♥✿
E = Q1
I 0
0 0
P1 , QB =
B˜1
B˜2
.
❑❤✐ ✤â✿ rank(B˜2) = n − rank(E).
✸✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ ✷✳✸ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
❜✮ ❍➺ ❝♦♥ ❝❤➟♠ ✈➔ ♥❤❛♥❤ ✷✳✸❛ ✈➔ ✷✳✸❜ ✤➲✉ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉
❦❤✐➸♥✳
❝✮ rank[B1 JB1 ... J n −1B1] = nf ✈➔ rank[B2 N B2 ... N v−1B2] = n∞
f
✶✾
❞✮ rank[λE − A B] = n ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C ✈➔ rank[E B] = n.
❡✮ rank[ αE − βA B ] = n ✈ỵ✐ t➜t ❝↔ (α, β) ∈ C2 \ (0, 0).
❈❤ù♥❣ ♠✐♥❤✳
✶✳ ❍➺ ❝♦♥ ❝❤➟♠ ❧➔ ♠ët ❖❉❊✱ ✈➟② ♥➯♥ ✤✐➲✉ ❦✐➺♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺
❝❤✉➞♥ ▲❚■ ✈➔ ✷✳✸❛ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ rank[ λ(I) − J B1 ] =
nf ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C
❍ì♥ ♥ú❛✱ t❛ ❝â
λI − J
rank[ λE − A B ] = rank[ λW ET − W AT W T B ] = rank
0
0
B1
λN − I B2
▼❛ tr➟♥ λN − I ❧➔ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✈ỵ✐ ♠å✐ ❤ú✉ ❤↕♥ λ ∈ ✈➔
rank[ λE − A B ] = n∞ + rank[ λI − J B1 ] = n
✈➻ ◆ ❧ô② ❧✐♥❤✱ ✈➻ ✈➟②
rank[ λI − N B2 ] = n∞
✈ỵ✐ ♠å✐ λ ∈ C ⇔ rank[ −N
B2 ] = rank[ N B2 ] = n∞ ,
✷✳ ✷❛ ⇐⇒ ✷❜ tø ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ ❤➺ ❝♦♥ ♥❤❛♥❤ ✷✳✸❜ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥
♥➳✉ t➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❧➔
Im[ B2 N B2 ... N v−1 B2 ] = Rn∞ ⇔ rank[ B2 N B2 ... N v−1 B2 ] = n∞
✷❜ ⇐⇒ ✷❝ ❍➺ (N, B2) ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ✭t÷ì♥❣ tü ❤➺ ❝❤✉➞♥ ▲❚■✮
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ rank[ λI − N B2 ] = n∞ ✈ỵ✐ ♠å✐ λ ∈ C. ✣✐➲✉ ❦✐➺♥ ♥➔② ❝è
✤à♥❤ ✈ỵ✐ ♠å✐ λ ∈ δ(N ) = {0} ✈➻ ◆ ❧ô② ❧✐♥❤✱ ✈➻ ✈➟②
rank[ λI − N B2 ] = n∞
⇐⇒
✈ỵ✐ ♠å✐ λ ∈ C
rank[ −N B2 ] = rank[ N B2 ] = n∞ .
✷❝ ⇐⇒ ✷❞ ❚❛ ❝â
rank[ E B ] = rank[ W ET W B ] = rank
❱➟②✱ rank[ N B2 ] = n∞ ⇐⇒ rank[ E
✷❞ ⇐⇒ ✷❡ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳
✷✵
Inf
0
B ] = n.
0
B1
N B2
= nf +rank[ N B2 ].
.
✸✳ ✸❛ ⇐⇒ ✸❝ ❈❤♦ ❤➺ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ✈➔ ❝❤♦ x1(0) = 0✳ ❑❤✐ ✤â✱ ✈ỵ✐
♠å✐ t1 > 0 w Rn tỗ t ởt ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ u = Cpv − 1 s❛♦
❝❤♦ x(t1) = w. ❱➟②
R˜0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] = Rn
⇐⇒ rank[ B1 JB1 ... J nf −1 B1 ] = nf
✈➔ rank[ B2 N B2 ... N n −1B2 ] = n∞.
▼➦t ❦❤→❝✱ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❤↕♥❣✱ ❦❤✐ ✤â t❛ ❜✐➳t
∞
Rx1 (0) = R˜0 +
x1
x2
|x1 = eJt x1 (0) ∈ Rnf , x2 = 0 ∈ Rn∞
= Rn
◆➯♥ ❤➺ ✭✷✳✸✮ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
✸❜ ⇐⇒ ✸❝ ❚❤❡♦ ✶ ✈➔ ✷✳
✸❜ ⇐⇒ ✸❞ ❚❤❡♦ ✶ ✈➔ ✷✳
✸❞ ⇐⇒ ✸❡ rank[ λE − A
α
βE
B ] = rank
−A B
= rank[ αE − βA B ]
❱➼ ❞ö ✷✳✹✳ ❳➨t ❤➺ s❛✉
x˙ 1 =
1 1
0 1
0 = x2 +
x1 +
−1
0
0
1
u
u.
❚❛ ❝â
rank[ B1 JB1 ] = rank
0 1
1 1
✈➔rank[ B2
=2
BN2 ] = rank
−1 0
0
0
= 1 < 2.
❱➟② ❤➺ tr➯♥ ❦❤æ♥❣ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✱ tr♦♥❣ ❦❤✐ ✤â ❤➺ t
ú ỵ ợ tr t tữợ ợ t tr ỵ
trữợ ổ ũ ủ t t sè✱ tø ❦❤❛✐ tr✐➸♥ ❝õ❛ ❤➺ tr♦♥❣ ✭❲❈❋✮ ❤❛②
❣✐→ trà r✐➯♥❣ ❝➛♥ t❤✐➳t✳ ▼ët ❝→❝❤ tèt ❤ì♥ ❧➔ t❤ỉ♥❣ q✉❛ ❞↕♥❣ ❜➟❝ t❤❛♥❣✳
❍➺ ♠æ t↔ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ ❧➔ ổ t õ t t ữủ ợ ồ tf > 0 ✈➔
x1 (0) ∈ R, w ∈ R, tỗ t ởt ữủ u Cpv1 s x(tf ) = w.
ỵ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣
✷✶
✶✳ ❍➺ ✭✷✳✸✮ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭❘✲❝♦♥tr♦❧❧❛❜❧❡✮
✷✳ ❍➺ ❝♦♥ ❝❤➟♠ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭❘✲❝♦♥tr♦❧❧❛❜❧❡✮
✸✳ rank[ λE − A B ] = n ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C
✹✳ rank[ B1 JB1 ... J n −1B1 ] = nf .
f
❈❤ù♥❣ ♠✐♥❤✳
✶ ⇐⇒ ✷ ❚ø ✤à♥❤ ♥❣❤➽❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ♥➳✉
R˜0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N v−1 B2 ]
= Rnf ⊕ Im[ B2 N B2 ... N v−1 B2 ]
❱➟②✱ Im[ B1 JB1 ... J n −1B1 ] = Rn ⇐⇒❍➺ ❝♦♥ ❝❤➟♠ ✭✷✳✸❛✮ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉
❦❤✐➸♥✳
✷ ⇐⇒ ✸ ❙✉② r❛ trü❝ t✐➳♣ tø ✷✳✷✳
❱➟②✱ ✤✐➲✉ ❦❤✐➸♥ ❤♦➔♥ t♦➔♥ ❜❛♦ ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝✳ ❈❤✐➲✉ ♥❣÷đ❝ ❧↕✐
❦❤ỉ♥❣ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳
f
f
❱➼ ❞ư ✷✳✺✳
✶✳ ❳➨t ❤➺ ð ✈➼ ❞ư ✭✷✳✹✮ ✤÷đ❝ ❝❤♦ ❜ð✐
x˙1 =
1 1
0 1
0 = x2 +
x1 +
−1
0
0
1
u
u.
❍➺ ❝♦♥ ❝❤➟♠ tr➯♥ ❧➔ ❈✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥❤÷ ✤➣ ❝â✱ ♥➯♥ ❤➺ tr➯♥ ❧➔ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳
✷✳ ❈❤♦ ◆ ❧ô② ❧✐♥❤ ✈➔ ①➨t ❤➺ N x˙ = x + Bu. tr ỗ ✈➔ ♥â
❧✉ỉ♥ ❧➔ ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳
❍➺ q✉↔ ✷✳✶✳ ❳➨t ❤➺ ✭✷✳✷✮ ✈ỵ✐ ❝❤ị♠ ♠❛ tr➟♥ ❝❤➼♥❤ q✉② λE − A✳ ❑❤✐ ✤â✱ ❤➺ tr➯♥ ❈✲
✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❤➺ ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ rank[ E
B ] = n.
ứ t q trữợ t t ❣✐→♥ ✤♦↕♥ tr♦♥❣ ♥❣❤✐➺♠ x(t) t↕✐
♠ët sè ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ I✳ ❚❤❡♦ ✤â✱ ✤➦t Dn = ∞(R, Rn) ❧➔ t➟♣ ✈ỉ ❤↕♥ ❝→❝
❤➔♠ ❦❤↔ ✈✐ ✈ỵ✐ ❣✐→ trà tr♦♥❣ Rn ✈➔ ❝♦♠♣❛❝t tr♦♥❣ R. ❈→❝ ♣❤➛♥ tû Dn ✤÷đ❝ ❣å✐ ❧➔
❝→❝ ❤➔♠ t✐➯✉ ❝❤✉➞♥✳
❈❤ù♥❣ ♠✐♥❤✳
✷✷
✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ▼ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ f : Dn → Rn ✈ỵ✐
f (α1 φ1 + α2 φ2 ) = α1 f (φ1 ) + α2 f (φ2 )
✈ỵ✐ ♠å✐ φ1, φ2 ∈ Dn, α1, α2 ∈ R
✣÷đ❝ ❣å✐ ❧➔ ❤➔♠ s✉② rë♥❣ ♥➳✉ ♥â ❧✐➯♥ tö❝✱ ✤â ❧➔ f (φ) → 0 tr♦♥❣ Rn ✈ỵ✐ ♠å✐ ❞➣②
(φi )i∈N ✈ỵ✐ φ1 → 0 tr♦♥❣ Dn ✳
❱➼ ❞ư ✷✳✻✳ P❤➙♥ ❜è ❉✐r❛❝ ❞❡❧t❛ δα ∈ C n ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ δα(φ) = φ(α) ✈ỵ✐ ♠å✐
φ ∈ Dn , α R
+
(x) =
0
x=
ú ỵ ∈ Dn ✈➔ tˆ > 0 ✤õ ❧ỵ♠ t❤➻ ❝è ✤à♥❤ r➡♥❣
tˆ
∞
ˆ
φ(t)dt
ˆ
φ(t)dt
=−
φ(0) = −(φ(tˆ) − φ(0)) = −φ(t)|t0 = −
0
0
ˆ
ˆ
H(t)φ(t)dt
=: H()
=
R
H(t) = 0 ợt < 0 ữợ ✤ì♥ ✈à✳ ❚❛ t➻♠ ✤÷đ❝ q✉❛♥ ❤➺ δ0 = H.˙ ❚❛
1 ✈ỵ✐t ≥ 0
❝ơ♥❣ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ sü ❞à❝❤ ❝❤✉②➸♥ ❝õ❛ H ❜➡♥❣ Hα(t) := H(t − α) ✈➔ δα = H˙ α
❍❛✐ ♣❤➙♥ ❜è f1, f2 ∈ C n ữ f1() = f2() ợ ồ φ ∈ Dn✳ ❚❤❡♦ ✤â✱
x : I → Rn , I R ữủ ỷ ỵ ữ ởt số ①→❝ ✤à♥❤ tr➯♥ R ❜ð✐ x(t) = 0 ✈ỵ✐ t ∈
/ I.
❚✉② ♥❤✐➯♥✱ ♥❣❤✐➺♠ ❜à ❤↕♥ ❝❤➳ t↕✐ ♠ët sè ✤➳♠ ✤÷đ❝ ❝→❝ ✤✐➸♠ τj ∈ T ⊆ R.
✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ●✐↔ sû t➟♣ T = {τj ∈ R|τj < τj+1 ∈ Z} ❦❤ỉ♥❣ ❝â ✤✐➸♠ tư ♥➔♦✳
▼ët ❤➔♠ s✉② rë♥❣ x ∈ C n ✤÷đ❝ ❣å✐ ❧➔ ✐♠♣✉❧s✐✈❡ s♠♦♦t❤ õ õ t ữủ t
ữợ
x = x + ximp ,
xˆ =
xˆj ,
✭✷✳✹✮
j∈Z
t↕✐ xˆj ∈ C ∞(|τj , τj+1|, Rn) ✈ỵ✐ ♠å✐ j ∈ Z ✈➔ ❜ë ♣❤➟♥ ①✉♥❣ ❧ü❝ ximp ❝â ❞↕♥❣
qj
(
ximp,j =
✭✷✳✺✮
cij ∈ Rn , qj ∈ N0 .
cij δτj i),
i=0
n (T).
❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♣❤➙♥ ❜è ①✉♥❣ ❧ü❝ trì♥ ✤÷đ❝ ❦➼ ❤✐➺✉ Cimp
n (C) ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❦❤❛✐ ❦❤✐➸♥ ✭✷✳✸✮✳
❇ê ✤➲ ✷✳✹✳ ✶✳ P❤➙♥ ❜è x ∈ Cimp
n (C) t❛ ❝â t❤➸ ❦➼ ❤✐➺✉ ❣✐→ trà ❤➔♠ sè x(t) ✈ỵ✐ ♠é✐ t ∈ R \ T ❜ð✐ x(t) = x
✷✳ ❱ỵ✐ x ∈ Cimp
ˆj
✈ỵ✐ t ∈ (τj , τj+1) ✈➔ lim x(τj−) = limt→τ xˆj−1(t) ✈➔ lim x(τj+) = limt→τ xˆj−1(t) ✈ỵ✐ ♠
−
j
+
j
✷✸
τj ∈ T.
n (T) ♥➡♠ tr♦♥❣ C n (T).
✸✳ ❚➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ ✈➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ x ∈ Cimp
imp
n
✹✳ ❚➟♣ Cimp(T) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦ ❜à ✤â♥❣ trữợ ợ A
C (R, Rm,n ).
n (C) t↕✐ τ ∈ T ✤÷đ❝ ❦➼ ❤✐➺✉ iord(x)|
✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ❇➟❝ ①✉♥❣ ❝õ❛ x ∈ Cimp
τ
j
j
:=
♥➳✉ x ❝â t❤➸ ❧✐➯♥ ❦➳t ✈ỵ✐ ❤➔♠ ❧✐➯♥ tư❝ tr♦♥❣ [τj−1; τ j + 1] ✈➔ q✱ ✈ỵ✐ 0 ≤ q ≤ ∞
❧➔ ♠ët sè ♥❣✉②➯♥ ✤õ ❧ỵ♥
−q − 2
x|[τj−1 ;τ j+1] ∈ C q ([τj−1 , τj+1 ], Rn ).
◆â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ iord(x)|τ ; = −1 ♥➳✉ ① ❝â t❤➸ ữủ t ợ ởt
tử tr [j1; j + 1] ♥❣♦➔✐ ✤✐➸♠ t = τj ✈➔ ♥â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛
j
iord(x)|τj := max{i ∈ N0 |0 ≤ i ≤ qj , cij = 0}
▼➦t ❦❤→❝✱ ❜➟❝ ①✉♥❣ ❝õ❛ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ iordx; = maxτ ∈Tiord(x)|
j
τj
.
n (T) ✈➔ A ∈ C ∞ (R, Rm,n ). ❑❤✐ ✤â iordAx ≤ iordx ✈ỵ✐ t➼♥❤
❇ê ✤➲ ✷✳✺✳ ❈❤♦ x ∈ Cimp
✤ì♥ ♥❤➜t m = n ✈➔ A(τj ) ❦❤↔ ♥❣❤à❝❤ ✈ỵ✐ ♠é✐ τj ∈ T.
❱➼ ❞ư ✷✳✼✳ ❳➨t ♠➝✉ sì ♠↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè s❛✉✿
❍➻♥❤ ✷✳✷✿ ▼↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥✳
x1 − x4 = u(t),
C(x˙ 1 − x˙ 2 ) +
1
(x3 − x2 ) = 0,
R
x3 = A(x4 − x2 ),
x4 = 0,
✷✹
✈ỵ✐ ✤➛✉ ✈➔♦ ✤✐➺♥ →♣ u(t) = 0 ✈ỵ✐t < 0,
1 ✈ỵ✐t ≥ 0
❱ỵ✐ x4 = 0, x1 = u(t), x˙ 1 = u˙ ✈➔ x3 = −Ax2 t❛ ✤÷đ❝
x˙ 2 = −
1
(A + 1)x2 + u.
˙
CR
❱➻ sü t❤❛② ✤ê✐ tr♦♥❣ ✤✐➺♥ →♣ ✤➛✉ ✈➔♦✱ ✉ ❦❤æ♥❣ ❦❤↔ ✈✐✳ ❈❤♦ u = H ✈➔ A → ∞✱
♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣
x1 − x4 = H,
C(x˙ 1 − x˙ 2 ) +
1
(x3 − x2 ) = 0,
R
x2 = 0,
x4 = 0,
✈ỵ✐ ♥❣❤✐➺♠ x1 = H, x2 = 0, x3 = −RC H˙ = −RCδ0✈➔ x4 = 0✳ ❍➺ tr➯♥ ❧➔ ♠ët ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè ✈ỵ✐ ❝❤➾ sè ν = 2 ✭❤♦➦❝ µ = 1✮ ✈➔ iordf = −1✳ Ð ♠é✐ ❣✐→ trà ♥❤➜t
q✉→♥ ❜❛♥ ✤➛✉✱ ✈➼ ❞ö x(−1) = 0 t❛ ❝â ♠ët ♥❣❤✐➺♠ ✤ì♥ x ✈ỵ✐ iordx = 0.
✣è✐ ợ trữớ ủ t ừ ữỡ tr ✤↕✐ sè ❜➜t ❜✐➳♥ ✤➲✉ ❝õ❛ ❞↕♥❣
n (T)✱ jordf = q ∈ Z ∪ {−∞} t❛ ❝â t❤➸ t✐➳♥ ❤➔♥❤ ữ s
E x = Ax + f ợ f Cimp
❚❤ù ♥❤➜t✱ t❛ ❝â t❤➸ ❜✐➳♥ ✤ê✐ ❝➦♣ ♠❛ tr➟♥ (E, A) t❤➔♥❤ (W CF )
(E, A) ∼ (W ET, W AT ) =
Inf
0
0
N
,
J
0
0 In∞
.
❱➟②✱ t❛ ❝â
x˙ = Jx1 + f1 ,
N x˙ 2 = x2 + f2 ,
✭✷✳✻❛✮
✭✷✳✻❜✮
❚↕✐ x1 = T −1x✳ ✣è✐ ✈ỵ✐ ♣❤➙♥ ❜è ❖❉❊ ✭✷✳✻❛✮ t❛ ❝â t❤➸ ①➨t ♥❣❤✐➯♠ ❝ì ❜↔♥ ❝õ❛
x2
♠❛ tr➟♥ X(t) t❤ä❛ ♠➣♥
˙
X(t)
= JX(t),
X(t0 ) = I,
n (T) s♦❧✈❡s ✭✷✳✻❛✮ ♥➳✉
♥❣❤➽❛ ❧➔ X(t) = eJ(t−t ) ∈ C ∞(R, Rn ,n )✳ ❱➟②✱ ♣❤➙♥ ❜è x˜ ∈ Cimp
n (T) s♦❧✈❡s
✈➔ ❝❤➾ ♥➳✉ z = X −1x˜ ∈ Cimp
0
f
f
z˙ = g1 = X −1 f1 ,
✷✺