Chéo hóa tương đẳng đồng thời hệ hai, ba ma trận trên trường số

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

✣➄◆● ❚❍➚ ❚❍❯ ❚❍❷❖

❈❍➆❖ ❍➶❆ ❚×❒◆● ✣➃◆●
✣➬◆● ❚❍❮■ ❍➏ ❍❆■✱ ❇❆ ▼❆
❚❘❾◆ ❚❘➊◆ ❚❘×❮◆● ❙➮

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵


❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆
✣➄◆● ❚❍➚ ❚❍❯ ❚❍❷❖

❈❍➆❖ ❍➶❆ ❚×❒◆● ✣➃◆●
✣➬◆● ❚❍❮■ ❍➏ ❍❆■✱ ❇❆ ▼❆
❚❘❾◆ ❚❘➊◆ ❚❘×❮◆● ❙➮
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ✈➔ ❧➼ t❤✉②➳t số
số

ữớ ữợ

▲➊ ❚❍❆◆❍ ❍■➌❯

❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵

✐

▼ư❝ ❧ư❝
▲í✐ ♠ð ✤➛✉

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶

✹

✷ ❈❤➨♦ ❤â❛ t÷ì♥❣ ✤➥♥❣ ỗ tớ tr
ố ự t❤ü❝
✶✶
✷✳✶

❍➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✶

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ❤➺ ❤❛✐ ♠❛ tr➟♥
✤è✐ ①ù♥❣ t❤ü❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✶✳✷
✷✳✷

✶✶

✶✶

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ♠ët ❤➺ ❤❛✐ ♠❛ tr➟♥
✤è✐ ①ù♥❣ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

❍➺ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✷✳✶

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ❤➺ ❜❛ ♠❛ tr➟♥
✤è✐ ①ù♥❣ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷✳✷

✷✸

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ♠ët ❤➺ ❜❛ ♠❛ tr➟♥ ✤è✐
①ù♥❣ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



õ tữỡ ỗ tớ ❤➺ ❤❛✐ ✈➔ ❜❛ ♠❛ tr➟♥
❍❡r♠✐t
✸✼
✸✳✶

❍➺ ❤❛✐ ♠❛ tr➟♥ ❍❡r♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼


✐✐

✸✳✶✳✶

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ❤➺ ❤❛✐ ♠❛ tr➟♥
❍❡r♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✶✳✷
✸✳✷

✸✼

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ♠ët ❤➺ ❤❛✐ ♠❛ tr➟♥
❍❡r♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

❍➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✸✳✷✳✶
✸✳✷✳✷

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ♠ët ❤➺ ❜❛ ♠❛

tr➟♥ ❍❡r♠✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ❤➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t

✹✾

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✵
✻✶


✶

▲í✐ ♠ð ✤➛✉
❚❛ ♥â✐ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣ A ❝➜♣ n tr➯♥ tr÷í♥❣ F ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣
✤÷ì♥❣ ✤÷đ❝ ♥➳✉ tỗ t tr ổ P n s❛♦ ❝❤♦

P −1 AP ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦❀ ♠❛ tr➟♥ ✈✉ỉ♥❣ A ❝➜♣ n tr➯♥ tr÷í♥❣ F
❧➔ ❝❤➨♦ ❤â❛ tữỡ ữủ tỗ t tr ổ P ❝➜♣ n ❦❤↔
♥❣❤à❝❤ s❛♦ ❝❤♦ P H AP ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❚❛ ❜✐➳t r➡♥❣ ♠å✐ ♠❛ tr➟♥
❍❡r♠✐t A Cnìn õ tữỡ ữủ tỗ t ởt
tr P Cnìn , P H P = In s❛♦ ❝❤♦ P H AP Rnìn
tr ữớ ✤➦t r❛ ❧➔✱ ✈ỵ✐ ♠ët ❤➺ ❝→❝ ♠❛
tr➟♥ ❍❡r♠✐t A1 , A2 , . . . , Am ✱ ❝â tỗ t ổ tr

P Cnìn õ ỗ tớ tt tr Ai i = 1, m

õ tỗ t↕✐ ❤❛② ❦❤æ♥❣ ♠ët ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥ P s❛♦ ❝❤♦

P H Ai P, ∀i = 1, m ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ
tỗ t tr P õ ỗ tớ ✤÷đ❝ ❤➺ tr➯♥ ❧➔ ❣➻❄
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝❤➾ t➟♣ tr✉♥❣ ❣✐↔✐ q✉②➳t ✈➜♥ ✤➲ ♥➯✉ tr➯♥
✤è✐ ✈ỵ✐ ❤➺ ❤❛✐ ❤♦➦❝ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✱ ♠❛ tr rt

õ tữỡ ỗ tớ tr
tr trữớ số ỗ ữỡ
ữỡ ởt sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët
▲✉➟♥ ✈➠♥ ✧

sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❧✐➯♥ q✉❛♥ ✤➳♥ ❝❤➨♦ ❤â❛ ♠❛ tr➟♥ tr➯♥ ♠ët tr÷í♥❣


✷

sè ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳

❈❤÷ì♥❣ ✷✳ ❈❤➨♦ ❤â❛ tữỡ ỗ tớ
tr ✤è✐ ①ù♥❣ t❤ü❝✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉
ừ õ tữỡ ỗ tớ ❤➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✱
❤➺ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✱ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤➨♦ ❤â❛ tữỡ
ỗ tớ ợ õ tữỡ ữỡ ỗ tớ

ữỡ õ tữỡ ỗ tớ ✈➔ ❜❛
♠❛ tr➟♥ ❍❡r♠✐t✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➜♥ ✤➲ t÷ì♥❣ tü ♥❤÷ tr♦♥❣
❈❤÷ì♥❣ ✷ ❝❤♦ ❝→❝ ❤å ♠❛ tr➟♥ rt
ữủ t ớ sỹ ữợ ❣✐ó♣ ✤ï t➟♥ t➻♥❤
❝õ❛ ❚❙✳ ▲➯ ❚❤❛♥❤ ❍✐➳✉✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥✳ ◆❤➙♥ ❞à♣ ♥➔②✱ tỉ✐

①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ✤➣ t➟♥ t➻♥❤
❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❈❤ó♥❣
tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ớ ỡ qỵ rữớ ❤å❝ ◗✉②
◆❤ì♥✱ P❤á♥❣ ✣➔♦ t↕♦ s❛✉ ✤↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ ũ qỵ t ổ
ợ ồ sè ✈➔ ❧➼ t❤✉②➳t sè ❦❤â❛ ✷✶ ✤➣ ❞➔② ❝æ♥❣ ❣✐↔♥❣
❞↕② tr♦♥❣ s✉èt ❦❤â❛ ❤å❝✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ ❝❤ó♥❣ tỉ✐ tr♦♥❣ q✉→
tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ◆❤➙♥ ✤➙②✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥
t❤➔♥❤ ❝↔♠ ì♥ sü ❤é trñ ✈➲ ♠➦t t✐♥❤ t❤➛♥ ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥
t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ✤➸ ❝❤ó♥❣ tỉ✐ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥
✈➠♥
ũ ữủ tỹ ợ sỹ ộ ❧ü❝ ❝è ❣➢♥❣ ❤➳t sù❝ ❝õ❛
❜↔♥ t❤➙♥✱ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔
❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣
t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ỳ õ ỵ ừ qỵ t ổ


✸

❣✐→♦ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳
❇➻♥❤ ✣à♥❤✱ ♥❣➔②

t❤→♥❣

♥➠♠ ✷✵✷✵

❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐

✣➦♥❣ ❚❤à ❚❤✉ ❚❤↔♦

✹

❈❤÷ì♥❣ ✶

▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ ♥➔②✱ F ❧➔ tr÷í♥❣ sè t❤ü❝ R ❤❛② tr÷í♥❣ sè ♣❤ù❝ C✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼❛ tr➟♥ t❤ü❝ ✈✉ỉ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣
t❤ü❝ ♥➳✉ AT = A✱ tr♦♥❣ ✤â AT ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr A
tr A Cnìn ữủ ồ ố ①ù♥❣ ♣❤ù❝ ♥➳✉ AT = A✳
▼❛ tr➟♥ ♣❤ù❝ ✈✉æ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❍❡r♠✐t ♥➳✉ A = AH ✱ tr♦♥❣
✤â AH ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❧✐➯♥ ❤ñ♣ ❝õ❛ ♠❛ tr➟♥ A✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼❛ tr➟♥ A ∈ Fnìn ữủ ồ õ tữỡ ữỡ
ữủ tỗ t↕✐ ♠❛ tr➟♥ P ∈ Fn×n ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ P −1 AP ❧➔ ♠❛
tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❑❤✐ ✤â t❛ ♥â✐ A ❧➔ DS ✭❞✐❛❣♦♥❛❧✐③❛❜❧❡ ✈✐❛ s✐♠✐❧❛r✐t②✮
tr➯♥ F✳
▼❛ tr➟♥ A Fnìn ữủ ồ õ tữỡ ữủ tỗ t
tr P Fnìn s ❝❤♦ P H AP ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳
❑❤✐ ✤â t❛ ♥â✐ A ❧➔ DC ✭❞✐❛❣♦♥❛❧✐③❛❜❧❡ ✈✐❛ ❝♦♥❣r✉❡♥❝❡✮ tr➯♥ F✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❛✮ ▼ët ❤å ♠❛ tr➟♥ A1, . . . , Am Fnìn ữủ ồ
SDC tr F ♥➳✉ ❝â ♠ët ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ P ∈ Fn×n s❛♦ ❝❤♦ P H Ai P
❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ✈ỵ✐ ♠å✐ i = 1, m✳


✺

❜✮ ▼ët ❤å ♠❛ tr➟♥ A1 , . . . , Am Fnìn ữủ ồ SDS tr F ♥➳✉ ❝â
♠ët ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ P ∈ Fn×n s❛♦ P 1 Ai P tr ữớ

ợ ♠å✐ i = 1, m✳

◆❤➟♥ ①➨t ✶✳✶✳ ▼ët ❤å ♠❛ tr➟♥ A1, . . . , An ❝â t❤➸✿
✰✮ ▼é✐ Ai , i = 1, m ❧➔ ♠ët ❝❤➨♦ ❤â❛ t÷ì♥❣ ✤➥♥❣ ✤÷đ❝ tr➯♥ F ♥❤÷♥❣
❦❤ỉ♥❣ ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣ ỗ tớ ữủ tr F
ữỡ tỹ ồ õ tữỡ ữỡ ỗ tớ

ử t




0 1
1 1
 , A2 = 
 ∈ R2×2 ⊆ C2×2 .
A1 = 
1 1
1 0

❑❤✐ ✤â {A1 , A2 } ❧➔ ổ õ tữỡ ỗ tớ ữủ tr R
tr ❦❤✐ ♠é✐ Ai , i = 1, 2✱ ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣ ✤➥♥❣ ✤÷đ❝ tr➯♥ R✳
❚❤➟t ✈➟②✱ t❛ ❝â A1 T = A1 , A2 T = A2 ♥➯♥ tø♥❣ ♠❛ tr➟♥ ❧➔ ❝❤➨♦ ❤â❛
trü❝ ❣✐❛♦ ✤÷đ❝ ❧➛♥ ❧÷đt ❜ð✐




√
√

1−
−1+
5
5
2
2
√
√
√
√
√
√
 10−2√5
 10−2√5
10−2 5 
10−2 5 
P1 = 
 , P2 = 
√
√ .
√ 2 √ √1− 5√
√ 2 √ √−1+ √5
10−2 5

10−2 5

10−2 5

10−2 5

❑❤✐ ✤â
√

P1 T A 1 P1 = 

5+1
2

0

0
√

− 5+1
2


 , P2 T A2 P2 =

√
− 5+1
 2



0

0
√

5+1
2


,

❧➔ ❤❛✐ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳
❚✉② ♥❤✐➯♥ A1 ✈➔ A2 ổ õ ỗ tớ ữủ t t õ

det A1 = −1 = det A2 ♥➯♥ A1 ❦❤↔ ♥❣❤à❝❤✱

 

1 −1
−1 1
=

A1 −1 = − 
−1 0
1 0


✻

✈➔









−1 1 1 1
0 −1

=
.
M = A1 −1 A2 = 
1 0 1 0
1 1

❚❛ ❝â

PM (λ) = 0 ⇔ PA1 −1 A2 (λ) = 0
⇔

−λ

−1

1

1−λ

=0

⇔ λ (λ − 1) + 1 = 0
⇔ λ2 − λ + 1 = 0

√
1±i 3
.
⇔ λ± =
2
❉♦ M ❝â ❤❛✐ ❣✐→ trà r✐➯♥❣ ♣❤➙♥ ❜✐➺t

x
❇➙② ❣✐í✱ ❣✐↔ sû ❝â ♠❛ tr➟♥ P = 
z

♥➯♥
 M ❝❤➨♦ ❤â❛ ✤÷đ❝✳
y
 ❦❤ỉ♥❣ s✉② ❜✐➳♥ s❛♦ ❝❤♦
t



α1 0
 = diag (α1 , α2 ) := D1 ,
P T A1 P = 
0 α2


β1 0
 = diag (β1 , β2 ) := D2 .
P T A2 P = 
0 β2

❉♦ det A1 = −1
 = detA2 ♥➯♥ α1 α2 = 0 = β1 β2 ✳ ❚ù❝ ❧➔ D1 , D2 ❦❤↔
β1
α1 0 

♥❣❤à❝❤✳ ❑❤✐ ✤â
= D1 −1 D2 = P −1 A1 −1 A2 P = P −1 M P ✳ ❙✉② r❛
β2
0 α2
β2
β1
✈➔
❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ M ✱ tù❝ ❧➔
α1
α2
√
√
βi
1+i 3
1−i 3
∈ {λ± } = λ+
, λ− =
.
αi
2
2


✼

❙✉② r❛ β1 = λ+ . α1 , β2 = λ− . α2 ∈ C✳ ❉♦ λ± ∈ C \ R ♥➯♥ P ∈ C2×2 \ R2×2 ✳
❈ư t❤➸✱ t❛ 
❝ât❤➸ ①→❝ ✤à♥❤
 ♠❛ tr➟♥ P ❧➔♠ ❝❤➨♦ ❤â❛ M ✤➲✉ ❝â
 t➜t ❝↔ ❝→❝

 xλ− yλ+
 | x, y ∈ C ✳
❞↕♥❣ P ∈ 

 −x −y
❘ã r➔♥❣ ♥➳✉ x ∈ R t❤➻ xλ− ∈ C✱ ✈➔ x ∈ C t❤➻ P ∈ C2×2 \ R2×2 . ❉♦ ✤â

P ∈ C2×2 \ R2×2 ✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä ❦❤ỉ♥❣ ❝â P ∈ R2×2 ❦❤↔ ♥❣❤à❝❤ ♠➔
P T A1 P, P T A2 P ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✱ ♠➦❝ ❞ò ♠é✐ ♠❛ tr➟♥ A1 , A2
✤➲✉ ❝❤➨♦ ❤â❛ trü❝ ❣✐❛♦ tr♦♥❣ R2×2 ✳

▼➺♥❤ ✤➲ ✶✳✶✳ ◆➳✉ A = diag (α1In , . . . , αk In ) ✈ỵ✐ αi = αj ∈ F ∀i =
1

k

j; n1 + . . . + nk = n ✈➔ AB = BA t❤➻ B = diag (B1 , . . . , Bk ) ợ Bi
Fni ìni , i = 1, k ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ B ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ✭t÷ì♥❣ ù♥❣ ♠❛

tr➟♥ ❍❡r♠✐t✮ t❤➻ ❝→❝ Bi ❝ơ♥❣ ✈➟②✳

▼➺♥❤ ✤➲ ✶✳✷✳ ▼å✐ ♠❛ tr➟♥ ❍❡r♠✐t A ✤➲✉ ❝❤➨♦ õ trỹ ữủ
tỗ t ởt tr ✉♥✐t❛ U ∈ Cn×n , U H U = In s

U H AU Rnìn tr ữớ ❝❤➨♦✳ ❍ì♥ ♥ú❛✱ ♥➳✉ A ∈ Rn×n t❤➻ U ❝â
t❤➸ ❝❤å♥ ❧➔ ♠❛ tr➟♥ trü❝ ❣✐❛♦ U ∈ Rn×n , U T U = In ✳
❈❤♦ A1 , . . . , Am ∈ Fn×n ✱ ❦➼ ❤✐➺✉
m

L(λ) =

λi Ai ,

λ = (λ1 , . . . , λm ) ∈ Rm ,

i=1

✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝❤ị♠ ♠❛ tr➟♥ s✐♥❤ ❜ð✐ A1 , . . . , Am ✳

▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ A1, . . . , Am ∈ Fn×n✳ tỗ t Rm s
det L() = 0✱ ❣✐↔ sû λ1 = 0 t❤➻ A1 , . . . , Am ❧➔ ❙❉❈ tr➯♥ F ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
L(λ), A2 , . . . , Am ❧➔ ❙❉❈ tr➯♥ F✳

❉♦ ♠➺♥❤ ✤➲ tr➯♥✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤➾ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣✿
✭✐✮ ❍å A1 , . . . , Am ❝â ♠ët ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤❀


✽

✭✐✐✮ ❑❤æ♥❣ ❝â ♠❛ tr➟♥ ♥➔♦ ❦❤↔ ♥❣❤à❝❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët ♠✐➲♥ ♥❣✉②➯♥ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥
ổ õ ữợ ừ

ử Z, Q, R, C ❧➔ ❝→❝ ♠✐➲♥ ♥❣✉②➯♥✳
n×n
n×n
❜✮ ❱➔♥❤ ❝→❝
tr➟♥ ✈✉ỉ♥❣
 ♠❛ 
  R  , C  ❦❤æ♥❣
 ❧➔ ♠ët
 ♠✐➲♥♥❣✉②➯♥✳
1 0 0 1
0 1
0 0
0 1 1 0

=
=
=

✳
❈❤➥♥❣ ❤↕♥✱ 
0 0 0 0
0 0
0 0
0 0 0 0

▼➺♥❤ ✤➲ ✶✳✹✳ ◆➳✉ F ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ t❤➻ F[x] = F [x1, . . . , xn] ❧➔
♠ët ♠✐➲♥ ♥❣✉②➯♥✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû F ❧➔ ♠ët tr÷í♥❣✳ ❳➨t F [x1 , . . . , xn ] ✈ỵ✐ ♠ët t❤ù tü
✤ì♥ t❤ù❝ ♥➔♦ ✤â✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ f, g ∈ F [x1 , . . . , xn ]✱ t❛ ✈✐➳t f, g t❤❡♦
t❤ù tü ✤ì♥ t❤ù❝ tr➯♥ ✤➙② ♥❤÷ s❛✉✿

f (x) = fα xα + . . . ,
g(x) = gβ xβ + . . . ,
tr♦♥❣ ✤â fα , gβ ∈ F \ {0} ❧➔ ❤➺ sè ❝❛♦ ♥❤➜t ❝õ❛ f (x) ✈➔ g(x) ù♥❣ ✈ỵ✐
t❤ù tü ✤ì♥ t❤ù❝ tr➯♥✳ ❑❤✐ ✤â✱ ♥➳✉ f (x)g(x) = 0 t❤➻ ❤➺ sè ❝❛♦ ♥❤➜t ❝õ❛

h(x) = f (x)g(x) ∈ F [x1 , . . . , xn ] ❧➔ fα gβ = 0✳ ❉♦ F ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥
♥➯♥ fα = 0 ❤♦➦❝ gβ = 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ fα , gβ ∈ F \ {0}✳ ❱➟②
F [x1 , . . . , xn ] ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ▼❛ tr➟♥ U ∈ Cn×n ✤÷đ❝ ❣å✐ ❧➔ ✉♥✐t❛ ♥➳✉ U H U = In✳
◆➳✉ U Rnìn t U ữủ ồ trỹ ♥➳✉ ♥â ❧➔ ✉♥✐t❛✱ ♥❣❤➽❛ ❧➔

U T U = In ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ i) ▼❛ tr➟♥ ❍❡r♠✐t A ∈ Hn ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ①→❝ ✤à♥❤
❞÷ì♥❣ ♥➳✉ xH Ax

0, ∀x ∈ Cn×1 ✳ ❚❛ ✈✐➳t A

0✳




ii) tr ố ự A Rnìn ữủ ồ ỷ ữỡ
xT Ax

0, x Rnì1 ✳ ❚❛ ✈✐➳t A

0✳

iii) ▼ët ♠❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✤÷đ❝ ❣å✐ ❧➔ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥➳✉
♥â ❦❤↔ ♥❣❤à❝❤✳ ✣➸ ❝❤➾ A ❧➔ ①→❝ ✤à♥❤ ❞÷ì♥❣ t❛ ✈✐➳t A

0✳

▼➺♥❤ ✤➲ ✶✳✺✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ❍❡r♠✐t ❤❛② ✤è✐ ①ù♥❣ t❤ü❝✳ ❈→❝
♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿

i) A

0❀

ii) ▼å✐ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ✤➲✉ ❧➔ sè t❤ü❝ ❦❤æ♥❣ ➙♠✳

▼➺♥❤ ✤➲ ✶✳✻✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ❍❡r♠✐t ❤❛② ✤è✐ ①ù♥❣ t❤ü❝✳ ❈→❝
♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿

i) A

0❀

ii) ▼å✐ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ✤➲✉ ❧➔ sè t❤ü❝ ❞÷ì♥❣✳

▼➺♥❤ ✤➲ ✶✳✼✳ ✭P❤➙♥ t➼❝❤ ❝ü❝ ❤❛② ♣❤➙♥ t➼❝❤ P♦❧❛r✮ ❈❤♦ A ❧➔ ♠ët ♠❛
tr➟♥ ✈✉æ♥❣✳ ❑❤✐ ✤â tỗ t ởt tr t U ởt tr ỷ
ữỡ P ũ ợ A s ❝❤♦ A = U P ✳
◆➳✉ A t❤ü❝ t❤➻ U ✈➔ P t❤ü❝✳
◆➳✉ A ❦❤↔ ♥❣❤à❝❤ t❤➻ P ❧➔ ①→❝ ữỡ (U, P )

t
ú ỵ r➡♥❣ ♥➳✉ →♣ ❞ö♥❣ ♣❤➙♥ t➼❝❤ P♦❧❛r ❝❤♦ AT t❤➻ t❛ ❝â

AT = V Q, V H V = I, Q

0.

❑❤✐ ✤â✱ A = QT V T := P U ✳

▼➺♥❤ ✤➲ ✶✳✽✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ♥û❛ ữỡ n
õ ổ tỗ t ởt ♠❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ B ❝ị♥❣ ❝➜♣ n s❛♦ ❝❤♦

B 2 = A✳


✶✵

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû A ❝â ♣❤➙♥ t➼❝❤ A = U H ΛU ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲
✶✳✷✱ U H U = In , Λ = diag (λ1 , . . . , λn ) , λi ∈ R ∀i = 1, n✳ ❉♦ A

λi

0 ♥➯♥

0, ∀i = 1, n t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✺✳ ✣➦t
√
√
B = U H Λ U, Λ := diag

❑❤✐ ✤â B

√
0 ✈➔ B 2 = U H Λ U

λ1 , . . . ,

λn

0.

√
U H Λ U = U H ΛU = A✳

▼➺♥❤ ✤➲ ✶✳✾✳ ▼❛ tr➟♥ ❍❡r♠✐t A ❧➔ ♠❛ tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣ ❦❤✐ ✈➔ ❝❤➾
❦❤✐ ♠å✐ ✤à♥❤ t❤ù❝ ❝♦♥ ❝❤➼♥❤ ❞➝♥ ✤➛✉ ❝õ❛ ♥â ✤➲✉ ❞÷ì♥❣✳
▼❛ tr➟♥ ❝♦♥ ❝❤➼♥❤ ❞➝♥ ✤➛✉ ❝➜♣ k ❝õ❛ ♠❛ tr➟♥ A ❝➜♣ n ❧➔ ♠❛ tr➟♥ ❧➜②
tr➯♥ k ❞á♥❣ ✤➛✉ ✈➔ k ❝ët ✤➛✉ ❝õ❛ A✳


1 2 3


 , t❛ ❝â✿
❱ỵ✐ A = 
4
5
6





7 8 9

❱➼ ❞ư ✶✳✹✳

▼❛ tr➟♥ ❝♦♥ ❝❤➼♥❤ ❞➝♥ ✤➛✉ ❝➜♣ ✶ ❧➔ 1 .


1 2
.
▼❛ tr➟♥ ❝♦♥ ❝❤➼♥❤ ❞➝♥ ✤➛✉ ❝➜♣ ✷ ❧➔ 
4 5




ữỡ

õ tữỡ ỗ tớ
❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔✴❤♦➦❝
✤õ ✤➸ ♠ët ❤å ❤❛✐ ❤♦➦❝ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ❧➔ õ tữỡ
ỗ tớ ữủ ở ữỡ ❝❤õ ②➳✉ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❬✷❪ ✈➔
❬✸❪✳

✷✳✶ ❍➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝
✷✳✶✳✶ ❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ❤➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✳

▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ A✱ B ∈ Rn×n ❧➔ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳
(i) ◆➳✉ A = In t❤➻ ❤➺ {In , B} ổ õ ỗ tớ ữủ

(ii) ợ A tr ố ự tũ ỵ {A, B} õ tữỡ
ỗ tớ (SDC) tỗ t tr P Rnìn
s P T AP ✈➔ P T BP ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳
❈❤ù♥❣ ♠✐♥❤✳ (i) ❉♦ B ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥➯♥ B õ trỹ
ữủ ự tỗ t tr➟♥ trü❝ ❣✐❛♦ P tù❝ ❧➔ P ∈ Rn×n , P T P = In
s❛♦ ❝❤♦ P T BP = D = diag(d1 , . . . , dn ).


✶✷

❍❛②



P T IP = I,


P T BP = D.

❱➟② {In , B} õ tữỡ ỗ tớ ữủ ❜ð✐ ♠❛ tr➟♥ P ✳

˜ = P T BP ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ A˜B
˜=B
˜ A˜✳
(ii) (⇐) ✣➦t A˜ = P T AP ✱ B
❱➻ A˜ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ tỗ t tr trỹ U s

= D ˜ = diag(α1 , . . . , αn ).
U T AU
A

❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû DA˜ = diag(α1 In1 , . . . , αk Ink ) ✈ỵ✐ αi =

αj ∈ R, ∀i = j ✈➔ n1 + . . . + nk = n✳
˜˜ = U T BU
˜=B
˜ A˜ ♥➯♥ A˜˜ = U T AU
˜ ✈➔ B
˜ ❣✐❛♦ ❤♦→♥✳ ❚❤➟t ✈➟②✱
❉♦ A˜B
˜˜ = U T AU
˜
A˜˜B

˜
U T BU

˜
= U T A˜BU
˜ AU
˜
= UTB
˜
= U T BU

˜
U T AU

˜˜ A.
˜˜
=B

˜˜ = diag(B , . . . , B ) t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ tr♦♥❣ ✤â B = B T ∈
❙✉② r❛ B
1
k
i
i
Rni ×ni , ∀i = 1, k ✳

❱ỵ✐ ♠é✐ i = 1, k ✱ t❤❡♦ (i)✱ ❤➺ (Ini , Bi ) ❧➔ ❝❤➨♦ ❤â❛ tữỡ ỗ
tớ ữủ ởt tr trỹ Qi ∈ Rni ×ni ✿ Qi T Qi = Ini ✈➔ Bi =

Qi Λi Qi T tr♦♥❣ ✤â Λi ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ✣➦t Q = diag (Q1 , . . . , Qk )✳


✶✸

❑❤✐ ✤â QT Q = diag(Q1 T Q1 , . . . , Qk T Qk ) = In ✳ ❙✉② r❛



QT Q = In ,




˜˜ = diag (α I , . . . , α I ) ,
QT AQ
1 n1
k nk






˜˜ = diag (Λ , . . . , Λ ) .
QT BQ
1
k
❙✉② r❛

˜˜ = QT U T AU
˜ Q = QT U T P T A (P U Q) = V T AV,
QT AQ
˜˜ = QT U T BU
˜ Q = QT U T P T B (P U Q) = V T BV
QT BQ
❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❱➟② ❤❛✐ ♠❛ tr➟♥ A✱ B ❧➔ ❝❤➨♦ õ tữỡ
ỗ tớ ữủ tr V = P U Q✳

(⇒) ●✐↔ sû A✱ B ❝❤➨♦ ❤â❛ t÷ì♥❣ ỗ tớ ữủ õ tỗ t
tr ♥❣❤à❝❤ P s❛♦ ❝❤♦ P T AP = DA ✈➔ P T BP = DB ❧➔ ❤❛✐ ♠❛
tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❉♦ DA DB = DB DA ♥➯♥ DA ✈➔ DB ❧➔ ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳
❍❛② P T AP ✈➔ P T BP ❧➔ ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳

✷✳✶✳✷ ❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ♠ët ❤➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝

▼➺♥❤ ✤➲ ✷✳✷✳ ❈❤♦ 0

= A, B ∈ Rn×n ❧➔ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❝â

dim (ker A ∩ ker B) = k ✳ ❑❤✐ ✤â k < n✳ ❍ì♥ ♥ú❛✱
(1) ❱ỵ✐ k = 0✱ ❝â 2 tr÷í♥❣ ❤đ♣✿
(i) ♥➳✉ det L (λ) = 0, ∀λ ∈ R2 t❤➻ {A, B} ❦❤æ♥❣
(ii) tỗ t R2 s det L (λ) = 0✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣
q✉→t✱ ❣✐↔ sû det A = 0✱ t❤➻ {A, B} ❧➔ ❙❉❈ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A−1 B ❧➔ ❙❉❙✳

(2) ❱ỵ✐ k

1✱ õ tỗ t tr P s ❝❤♦




0k 0
0 0
 , P T BP =  k
,
P T AP = 
˜
0 A˜
0 B


✶✹

˜ B
˜ ∈ R(n−k)×(n−k) ✤è✐ ①ù♥❣ ✈➔ dim ker A˜ ∩ ker B
˜ = 0✳
✈ỵ✐ A,
˜ B

˜ ❧➔ ❙❉❈✳
❍ì♥ ♥ú❛✱ A, B ❧➔ ❙❉❈ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A,
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû k = n✳ ❑❤✐ ✤â dim (ker A ∩ ker B) = n✳ ❉♦

dim (ker A ∩ ker B)

dim (ker A) = n − rankA

♥➯♥ rankA = 0✳ ❉♦ ✤â A = 0✱ ♠➙✉ t❤✉➝♥✳ ❱➟② k < n✳

(1) ●✐↔ sû k = 0✱
(i) ●✐↔ sû det L (λ) = 0, ∀λ ∈ R2 ✈➔ A, B ❧➔ õ tỗ t
tr P s ❝❤♦ P T AP = DA ✈➔ P T BP = DB tr
ữớ tr Rnìn ợ



DA = diag (α1 , . . . , αn ) ,


DB = diag (β1 , . . . , βn ) ,

tr♦♥❣ ✤â αi , βi ∈ R, ∀i = 1, n✳ ❙✉② r❛

det (L (λ)) = det (λ1 A + λ2 B)
= det (P −1 )T (λ1 DA + λ2 DB ) P −1
= det2 P −1 . det (λ1 DA + λ2 DB )
= det2 P −1 . (λ1 α1 + λ2 β1 ) (λ1 α2 + λ2 β2 ) . . . (λ1 αn + λ2 βn )
n
2

= det P

−1

.

(λ1 αi + λ2 βi ) .
i=1

❉♦ det L (λ) = 0, ∀λ ∈ R2 ♥➯♥ ✤❛ t❤ù❝ t❤ü❝
n

f (x1 , x2 ) =

(αi x1 + βi x2 )
i=1
n

qi (x1 , x2 ), qi (x1 , x2 ) = αi x1 + βi x2 ∈ R [x1 , x2 ]

=
i=1


✶✺

❧➔ ✤❛ t❤ù❝ ❦❤æ♥❣✳ ❉♦ R [x1 , x2 ] tỗ t i {1, . . . , n}
s❛♦ ❝❤♦

0 ≡ qi (x1 , x2 ) = αi x1 + βi x2 , ∀ (x1 , x2 ) ∈ R2 .
❍❛② (αi , βi ) = (0, 0)✳ ❙✉② r❛

α
 1

✳✳✳




αi−1


AP ei = P DA ei = P 
0



αi+1



✳✳✳













 ei = 0.









αn
❉♦ ✤â AP ei = 0❀ ❤❛② P ei ∈ ker A. ❚÷ì♥❣ tü✱ P ei ∈ ker B. ❙✉② r❛

P ei ∈ (ker A ∩ ker B)✱ tr♦♥❣ ✤â ei ❧➔ ❝ët tå❛ ✤ë ❝õ❛ ✈❡❝ tì ✤ì♥ ✈à t❤ù i
tr♦♥❣ Rn ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t 0 = k = dim (ker A ∩ ker B)✳
❱➟② A, B ❦❤æ♥❣ ❙❉❈✳

(ii) ●✐↔ sû det A = 0✳
(⇒) ●✐↔ sû A, B õ tỗ t tr P ❦❤↔ ♥❣❤à❝❤ s❛♦
❝❤♦ P T AP = DA ✈➔ P T BP = DB ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ tr♦♥❣ Rn×n ✳
❙✉② r❛

P −1 A−1 BP = P −1 A−1 P −T

P T BP = DA −1 DB

❧➔ ✤÷í♥❣ ❝❤➨♦✳ õ A1 B

() sỷ tỗ t Q ∈ Rn×n ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ Q−1 A−1 BQ =
D Rnìn tr ữớ ổ t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû
D = diag (α1 In1 , . . . , αk Ink ) ✈ỵ✐ αi = αj ∈ R, ∀i = j ✈➔ n1 + . . . + nk = n✳


✶✻

❑❤✐ ✤â
−1

D = Q−1 A−1 BQ = Q−1 A−1 Q−T QT BQ = QT AQ
❙✉② r❛

QT AQ D = QT BQ
= QT BQ
=

T

QT AQ D

= DT QT AQ

T

T

= D QT AQ .
❚❤❡♦ ▼➺♥❤ ✤➲ 1.1 t❛ ❝â

QT AQ = diag (A1 , . . . , Ak )
✈ỵ✐ Ai ∈ Rni ×ni ✤è✐ ①ù♥❣✱ ∀i = 1, k. ❚ø ✤â✱


α1 In1
 A1


✳✳✳

QT BQ = 




αk Ink



✳✳✳

Ak








= diag (α1 A1 , . . . , αk Ak ) .
❉♦ Ai = Ai T i = 1, k tỗ t Qi ∈ Rni ×ni s❛♦ ❝❤♦



Qi T Qi = Ini ,


Ai = Qi T Λi Qi , tr♦♥❣ ✤â Λi ❧➔ ữớ ,

ợ ồ i = 1, k õ

QT BQ .


✶✼



T
Q1 Λ1 Q1

✳✳✳
QT AQ = 




Q1

=



T


Λ1

˜
=Q



✳✳✳

Qk T








Qk T Λk Qk


 Λ 1

✳✳✳




Λk


 Q1

✳✳✳






Qk









✳✳✳

Λk


 T
˜ ,
Q



tr♦♥❣ ✤â

T
Q1

✳✳✳
˜=
Q





Qk T



.



❚÷ì♥❣ tü✱

α1 Λ1

˜
QT BQ = Q





✳✳✳

αk Λ k


 T
˜ .
Q



❙✉② r❛

˜ T QT AQQ
˜ = diag (Λ1 , . . . , Λk ) ,
Q
˜ T QT BQQ
˜ = diag (α1 Λ1 , . . . , αk Λk ) .

Q
˜✳
❱➟② A ✈➔ B ❧➔ ❙❉❈ ✤÷đ❝ ❜ð✐ ♠❛ tr➟♥ P = QQ
(2) ●✐↔ sû (ker A ∩ ker B) = k > 0. ❉♦ Rn ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝ tì


✶✽

❊✉❝❧✐❞ ♥➯♥ t❛ ❝â t❤➸ ❧➜② {u1 , . . . , uk , uk+1 , . . . , un } ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥
❝õ❛ Rn s❛♦ ❝❤♦ {u1 , . . . , uk } ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ❝õ❛ (ker A ∩ ker B)✳
✣➦t U = ([u1 ]e , . . . , [un ]e ) = (u1 , . . . , un ) ❧➔ ♠❛ tr➟♥ ✤ê✐ ❝ì sð tø (e)
s❛♥❣ (u)✳ ❑❤✐ ✤â U T U = In ✳ ❍ì♥ ♥ú❛✱
 
T
u1 
 
uT 
 2
T
U AU =   A u1 , u2 , . . . , un
 ✳✳✳ 
 
 
uTn

= uTi Auj , ∀i, j = 1, n


0k 0
,

=
˜
0 A
✈➔


T



u1 
 
uT 
 2
T
U BU =   B u1 , u2 , . . . , un
 ✳✳✳ 
 
 
uTn

= uTi Buj , ∀i, j = 1, n


0k 0
,
=
˜
0 B
˜ B

˜ ∈ R(n−k)×(n−k) ❧➔ ố ự
tr õ A,
> 0 t tỗ t 0 = x˜ ∈ ker A˜ ∩ ker B
˜
◆➳✉ dim ker A˜ ∩ ker B
 
0k
Rn−k ✳ ❙✉② r❛ 0 = x =   ∈ Rn×1 ✳ ❑❤✐ ✤â
x˜

⊆


✶✾



U T AU x = 

0k 0



 x = 0,
0 A˜


0k 0
 x = 0.
U T BU x = 

˜
0 B

❙✉② r❛

0 = y = U x ∈ ker A,
0 = y = U x ∈ ker B.
❉♦ ✤â y ∈ (ker A ∩ ker B)✳ ❍ì♥ ♥ú❛✱ ∀j = 1, k, t❛ ❝â

uTj U x = uTj y
= uTj . u1 , . . . , uk , uk+1 , . . . , un . x
= uTj u1 , . . . , uTj uk , uTj uk+1 , . . . , uTj un . x
 
0
✳
 ✳✳ 
j
 
∨
= 0, . . . , 0, , , 0, . . . , 0  
1
0
 
 
x˜
= 0.
❙✉② r❛ y ⊥ (ker A ∩ ker B)✳ ❚ø ✤â s✉② r❛ y = 0✱ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â

˜ = 0.
dim ker A˜ ∩ ker B

❍ì♥ ♥ú❛

˜ B
˜ ❧➔ ❙❉❈
A,
˜ ❧➔ ❙❉❙
⇔ A˜−1 B


✷✵

⇔
⇔

−1 


0 0
0 0
  k
 ❧➔ ❙❉❙
 k
˜
˜
0 B
0 A

P T AP

−1

P T BP

❧➔ ❙❉❙

⇔ P −1 A−1 BP ❧➔ ❙❉❙
⇔ A−1 B ❧➔ ❙❉❙
⇔ A, B ❧➔ ❙❉❈.

❉ò♥❣ ♣❤➙♥ t➼❝❤ P♦❧❛r ð ▼➺♥❤ ✤➲ ✶✳✼✱ t❛ ❝â t❤➯♠ ♠ët sè ✤✐➲✉ ❦✐➺♥
t÷ì♥❣ ✤÷ì♥❣ ✤➸ ♠ët ❤å ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ❧➔ SDC ♥❤÷ s❛✉✳

▼➺♥❤ ✤➲ ✷✳✸✳ ❈❤♦ A, B ∈ Rn×n ❧➔ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥
s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿

(i) {A, B} ❝❤➨♦ õ tữỡ ỗ tớ ữủ
(ii) ỗ t ởt tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣ X s❛♦ ❝❤♦
AXB = BXA.
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♥➳✉ ❝❤å♥ Q =

√

X

0 ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲ ✶✳✽

t❤➻ ♠å✐ ♠❛ tr➟♥ ❝â ❞↕♥❣ P = U Q✱ ợ U tr trỹ tũ ỵ
õ tữỡ ỗ tớ tr A ✈➔ B ✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✱ A, B õ tữỡ ỗ tớ
ữủ ❝â P ∈ Rn×n ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ P T AP ✈➔ P T BP

❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ⑩♣ ❞ö♥❣ ♣❤➙♥ t➼❝❤ P♦❧❛r ❝❤♦ P t❛ ❝â

P = QU, U T U = I, QT = Q

0.


✷✶

❑❤✐ ✤â

P T AP

P T BP = P T BP

⇔ U T QT AQU

P T AP

U T QT AQU = U T QT BQU

U T QT AQU

⇔ AQQT B = BQQT A
⇔ AQ2 B = BQ2 A.
❑❤✐ ✤â X = Q2

❱➼ ❞ö ✷✳✶✳

0 t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ▼➺♥❤ ✤➲✳





0 1
1 1
,B = 
.
❛✮ ❳➨t A = 
1 1
1 0

⑩♣
 ử
t tr sỹ tỗ t ❝õ❛ ♠ët ♠❛ tr➟♥ X =
x y

 0 t❤ä❛ ♠➣♥ AXB = BXA✳
y z
❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✾✱

X

0⇔




x > 0,



xz − y 2 > 0.

▼➦t ❦❤→❝✱

AXB = BXA



 



0 1 x y 1 1
1 1 x y 0 1


=



⇔
1 1 y z
1 0
1 0 y z
1 1

 

y+z

y
y + z x + 2y + z
=

⇔
x + 2y + z x + y
y
x+y
⇔ x + 2y + z = y
⇔ x + y + z = 0.