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 sè ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✹ ▲❯❾◆ ữớ ữợ ❚❍❆◆❍ ❍■➌❯ ❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵ ✐ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✹ ✷ ❈❤➨♦ õ tữỡ ỗ tớ ♠❛ 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❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❍➺ ❜❛ ♠❛ 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 õ tữỡ ữỡ ữủ tỗ t tr➟♥ ✈✉æ♥❣ P ❝➜♣ n ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ P −1 AP ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦❀ ♠❛ tr➟♥ ✈✉ỉ♥❣ A ❝➜♣ n tr➯♥ tr÷í♥❣ F ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣ 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 Cn×n , P H P = In s❛♦ ❝❤♦ P H AP Rnìn tr ữớ ♥❤✐➯♥ ✈➜♥ ✤➲ ✤➦t r❛ ❧➔✱ ✈ỵ✐ ♠ët ❤➺ ❝→❝ ♠❛ tr➟♥ ❍❡r♠✐t A1 , A2 , , Am õ tỗ t ổ tr P Cnìn õ ỗ tớ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ Ai ✱ i = 1, m õ tỗ t ổ ởt ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥ P s❛♦ ❝❤♦ P H Ai P, ∀i = 1, m ❧➔ ❝→❝ ♠❛ tr➟♥ ữớ tỗ t tr P õ ỗ tớ ữủ 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 ❈❤÷ì♥❣ ✶✳ ▼ët 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♥❣ õ tữỡ ỗ tớ ✈➔ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❝❤➨♦ õ tữỡ ỗ tớ tr ố ①ù♥❣ t❤ü❝✱ ❤➺ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✱ ✈➔ ố ỳ õ 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 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❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ú tổ ụ ỷ ỡ qỵ ❇❛♥ ❧➣♥❤ ✤↕♦ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥✱ P❤á♥❣ ✣➔♦ t s ũ qỵ t ổ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❝❛♦ ❤å❝ ✣↕✐ sè ✈➔ ❧➼ t❤✉②➳t sè ❦❤â❛ ✷✶ ✤➣ ❞➔② ❝æ♥❣ ❣✐↔♥❣ ❞↕② tr♦♥❣ s✉èt ❦❤â❛ ❤å❝✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ ❝❤ó♥❣ tæ✐ 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❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ị ❧✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ✈ỵ✐ sü ♥é ❧ü❝ ❝è ❣➢♥❣ ❤➳t sù❝ ❝õ❛ ❜↔♥ t❤➙♥✱ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ú tổ rt ữủ ỳ õ ỵ qỵ t ổ ữủ 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❝ ✈✉æ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ♥➳✉ AT = A✱ tr♦♥❣ ✤â AT ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A✳ ▼❛ tr➟♥ A ∈ Cn×n ✤÷đ❝ ❣å✐ ❧➔ ✤è✐ ①ù♥❣ ♣❤ù❝ ♥➳✉ AT = A✳ ▼❛ tr➟♥ ♣❤ù❝ ✈✉ỉ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❍❡r♠✐t ♥➳✉ A = AH ✱ tr♦♥❣ ✤â AH ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❧✐➯♥ ❤ñ♣ ❝õ❛ ♠❛ tr➟♥ A✳ tr A Fnìn ữủ 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 õ tữỡ ữủ tỗ t tr P Fnìn ♥❣❤à❝❤ s❛♦ ❝❤♦ P H AP ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❑❤✐ ✤â t❛ ♥â✐ A ❧➔ DC ✭❞✐❛❣♦♥❛❧✐③❛❜❧❡ ✈✐❛ ❝♦♥❣r✉❡♥❝❡✮ tr➯♥ F✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❛✮ ▼ët ❤å ♠❛ tr A1, , Am Fnìn ữủ ❣å✐ ❧➔ SDC tr➯♥ F ♥➳✉ ❝â ♠ët ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ P ∈ Fn×n s❛♦ ❝❤♦ P H Ai P tr ữớ ợ i = 1, m✳ ✺ ❜✮ ▼ët ❤å ♠❛ tr➟♥ A1 , , Am Fnìn ữủ SDS tr➯♥ F ♥➳✉ ❝â ♠ët ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ P ∈ Fn×n s❛♦ ❝❤♦ P −1 Ai P ❧➔ tr ữớ ợ i = 1, m ◆❤➟♥ ①➨t ✶✳✶✳ ▼ët ❤å ♠❛ tr➟♥ A1, , An ❝â t❤➸✿ ✰✮ ▼é✐ Ai , i = 1, m ❧➔ ♠ët ❝❤➨♦ ❤â❛ t÷ì♥❣ ✤➥♥❣ ✤÷đ❝ tr F ữ ổ õ tữỡ ỗ t❤í✐ ✤÷đ❝ tr➯♥ F✳ ✰✮ ❚÷ì♥❣ tü ❝❤♦ ❤å ❝❤➨♦ õ tữỡ ữỡ ỗ tớ t    1  , A2 =   ∈ R2×2 ⊆ C2×2 A1 =  1 ❑❤✐ ✤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 2 √ √ √ √ √ √  10−2√5  10−2√5 10−2  10−2  P1 =   , P2 =  √ √  √ √ √1− 5√ √ √ √−1+ √5 10−2 10−2 10−2 10−2 ❑❤✐ ✤â √ P1 T A P1 =  5+1 0 √ − 5+1   , P2 T A2 P2 = √ − 5+1   0 √ 5+1  , ❧➔ ❤❛✐ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❚✉② ♥❤✐➯♥ A1 A2 ổ õ ỗ tớ ữủ t t❛ ❝â det A1 = −1 = det A2 ♥➯♥ A1 ❦❤↔ ♥❣❤à❝❤✱     −1 −1 =  A1 −1 = −  −1 ✻ ✈➔      −1 1 −1  =  M = A1 −1 A2 =  1 1 ❚❛ ❝â PM (λ) = ⇔ PA1 −1 A2 (λ) = ⇔ −λ −1 1−λ =0 ⇔ λ (λ − 1) + = ⇔ λ2 − λ + = √ 1±i ⇔ λ± = ❉♦ M ❝â ❤❛✐ ❣✐→ trà r✐➯♥❣ ♣❤➙♥ ❜✐➺t  x ❇➙② ❣✐í✱ ❣✐↔ sû ❝â ♠❛ tr➟♥ P =  z ♥➯♥  M ❝❤➨♦ ❤â❛ ✤÷đ❝✳ y  ❦❤ỉ♥❣ s✉② ❜✐➳♥ s❛♦ ❝❤♦ t   α1  = diag (α1 , α2 ) := D1 , P T A1 P =  α2   β1  = diag (β1 , β2 ) := D2 P T A2 P =  β2 ❉♦ det A1 = −1  = detA2 ♥➯♥ α1 α2 = = β1 β2 ✳ ❚ù❝ ❧➔ D1 , D2 ❦❤↔ β1 α1   ♥❣❤à❝❤✳ ❑❤✐ ✤â = D1 −1 D2 = P −1 A1 −1 A2 P = P −1 M P ✳ ❙✉② r❛ β2 α2 β2 β1 ✈➔ ❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ M ✱ tù❝ ❧➔ α1 α2 √ √ βi 1+i 1−i ∈ {λ± } = λ+ , λ− = αi 2 ✹✼ ⇔ P −1 A−1 BP ❧➔ ❙❉❙ ⇔ A−1 B ❧➔ ❙❉❙ ⇔ A, B ❧➔ ❙❉❈ ✸✳✷ ❍➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t ✸✳✷✳✶ ❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ♠ët ❤➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t ▼➺♥❤ ✤➲ ✸✳✹✳ ❈❤♦ A, B, C ∈ Cn×n ❧➔ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t✳ (i) ◆➳✉ A = In t❤➻ {In , B, C} õ tữỡ ỗ tớ ữủ tỗ t tr ✉♥✐t❛ P s❛♦ ❝❤♦ P H BP ✈➔ P H CP ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ (ii) ❱ỵ✐ A ❧➔ ♠❛ tr➟♥ ❍❡r♠✐t ✱ ❤➺ {A, B, C} ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣ ỗ tớ ữủ tỗ t ♠❛ tr➟♥ ✉♥✐t❛ P s❛♦ ❝❤♦ P H AP ✱ P H BP ✱ P H CP ✤æ✐ ♠ët ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ❱ỵ✐ A = In ✱ t❛ ①➨t ❤➺ (In , B, C)✳ (⇒) ●✐↔ sû (In , B, C) õ tữỡ ỗ tớ ữủ õ tỗ t tr t P s❛♦ ❝❤♦ P H In P = In , P H BP = DB ✈➔ P H CP = DC ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❉♦ DB DC = DC DB ♥➯♥ DB ✈➔ DC ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ❍❛② P H BP ✈➔ P H CP ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ () sỷ tỗ t tr t P tọ ♠➣♥ P H BP ✈➔ P H CP ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✱ B ✈➔ C ❝❤➨♦ ❤â❛ tữỡ ỗ tớ ữủ ợ õ tỗ t tr t U s  U H U = In ,     U H BU = DB ,      U H CU = D , C ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❱➟② ❤➺ {In , B, C} ❧➔ SDC ✳ ˜ = P H BP ✱ C˜ = P H CP ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ (ii) (⇒) ✣➦t A˜ = P H AP ✱ B ˜ B, ˜ C˜ ✤æ✐ ♠ët ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ❱➻ A˜ tr rt tỗ t õ A, t ♠❛ tr➟♥ ✉♥✐t❛ U s❛♦ ❝❤♦ ˜ = D ˜ = diag(α1 , , αn ) U H 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 H BU ˜=B ˜ A˜ ♥➯♥ A˜˜ = U H AU ˜ ✈➔ B ˜ ❣✐❛♦ ❤♦→♥✳ ❚❤➟t ✈➟②✱ ❉♦ A˜B ˜˜ = U H AU ˜ A˜˜B ˜ U H BU ˜ = U H A˜BU ˜ AU ˜ = UHB ˜ = U H BU ˜ U H AU ˜˜ A ˜˜ =B ˜ = U H AU ˜ ✈➔ C˜˜ = U H CU ˜ ❣✐❛♦ ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â A˜C˜ = C˜ A˜ ♥➯♥ A˜ ❤♦→♥✳ ˜˜ = diag(B , , B ) ✈➔ C˜˜ = diag(C , , C ) ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ s✉② r❛ B k k tr♦♥❣ ✤â Bi = Bi H ∈ Rni ×ni , Ci = Ci H ∈ Rni ×ni , ∀i = 1, k ✳ ˜˜ ✈➔ C˜˜ ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ❉♦ ✤â B ✈➔ C ˜ C˜ = C˜ B ˜ ♥➯♥ B ❍ì♥ ♥ú❛✱ ❞♦ B i i ❣✐❛♦ ❤♦→♥ ♥❤❛✉ ∀i = 1, k ✹✾ ❱ỵ✐ ♠é✐ i = 1, k ✱ t❤❡♦ (i)✱ ❤➺ (Ini , Bi , Ci ) õ tữỡ ỗ tớ ữủ ởt ♠❛ tr➟♥ trü❝ ❣✐❛♦ Qi ∈ Rni ×ni ✿ Qi H Qi = Ini ✱ Bi = Qi Λi Qi H ✈➔ Ci = Qi Γi Qi H tr♦♥❣ ✤â Λi , Γi ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ✣➦t Q = diag (Q1 , , Qk )✳ ❑❤✐ ✤â QH Q = diag(Q1 H Q1 , , Qk H Qk ) = In ✳ ❙✉② r❛    QH Q = In ,        ˜˜ = diag (α I , , α I ) , QH AQ n1 k nk  ˜˜ = diag (Λ , , Λ )   QH BQ k       ˜˜ = diag (Γ , , Γ ) QH CQ k ❙✉② r❛ ˜˜ = QH U H AU ˜ Q = QH U H P H A (P U Q) = V H AV, QH AQ ˜˜ = QH U H BU ˜ Q = QH U H P H B (P U Q) = V H BV QH BQ ˜˜ = QH U H CU ˜ Q = QH U H P H C (P U Q) = V H CV QH CQ ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❱➟② ❜❛ ♠❛ tr A, B, C õ tữỡ ỗ t❤í✐ ✤÷đ❝ ❜ð✐ ♠❛ tr➟♥ V = P U Q✳ ✸✳✷✳✷ ❚➼♥❤ ❙❉❈ ✈➔ t➼♥❤ ❙❉❙ ❝õ❛ ❤➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t ▼➺♥❤ ✤➲ ✸✳✺✳ ❈❤♦ A, B, C ∈ Cn×n ❧➔ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t t❤ä❛ ♠➣♥ dim (ker A ∩ ker B ∩ ker C) = k ✳ ❑❤✐ ✤â k < n✳ ❍ì♥ ♥ú❛✱ (1) ❱ỵ✐ k = 0✱ ❝â tr÷í♥❣ ❤đ♣ (i) ♥➳✉ det L (λ) = 0, ∀λ ∈ R3 t❤➻ A✱ B ✱ C ổ (ii) tỗ t R3 s❛♦ ❝❤♦ det L (λ) = 0✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû det A = 0✱ t❤➻ {A, B, C} ❧➔ ❙❉❈ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠é✐ A−1 B A1 C tữỡ ữỡ ợ ởt tr➟♥ ✤÷í♥❣ ❝❤➨♦ t❤ü❝✱ ✈➔ ❤➺ A−1 B, A−1 C SDS tr Rnìn (2) ợ k õ tỗ t tr t P s      0k 0 0  , P H BP =  k  , P H CP =  k , P H AP =  ˜ A˜ B C˜ ˜ B, ˜ C˜ ∈ C(n−k)×(n−k) ❧➔ ❍❡r♠✐t ✈➔ dim ker A˜ ∩ ker B ˜ ∩ ker C˜ ✈ỵ✐ A, = 0✳ ˜ B, ˜ C˜ ❧➔ ❙❉❈✳ ❍ì♥ ♥ú❛✱ A, B, C ❧➔ ❙❉❈ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A, ❈❤ù♥❣ ♠✐♥❤✳ ❚÷ì♥❣ tü ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ▼➺♥❤ ✤➲ ✷✳✻ t❛ ❝â k < n (1) ●✐↔ sû ❦ ❂ ✵✳ (i) ●✐↔ sû t❤➯♠ det L (λ) = 0, ∀λ ∈ R3 ✈➔ A, B, C ❧➔ ❙❉❈✳ õ tỗ t tr P Cn×n s❛♦ ❝❤♦ P H AP = DA ✱ P H BP = DB ✈➔ P H CP = DC tr ữớ ợ DA = diag (α1 , , αk ) ,     DB = diag (β1 , , βk ) ,      D = diag (γ , , ) C k ú ỵ r DA H = P H AP H = P H AH P = P H AP = DA , ✈➔ t÷ì♥❣ tü DB H = DB ✱ DC H = DC ♥➯♥ αi , βi , γi ∈ R, ∀i = 1, n✳ ❙✉② r❛ det L(λ) = det(λ1 A + λ2 B + λ3 C) = det (P −1 )H (λ1 DA + λ2 DB + λ3 DC ) P −1 = det P −1 = det P −1 det (λ1 DA + λ2 DB + λ3 DC ) [(λ1 α1 + λ2 β1 + λ3 γ1 ) (λ1 αn + λ2 βn + λ3 γn )] ✺✶ n = det P −1 (λ1 αi + λ2 βi + λ3 γi ) i=1 ❧➔ ♠ët ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè t❤ü❝✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❣✐↔ t❤✐➳t✱ ❞♦ det L (λ) = 0, ∀λ ∈ R3 ♥➯♥ ✤❛ t❤ù❝ n f (x1 , x2 , x3 ) = (λ1 αi + λ2 βi + λ3 γi ) i=1 ❧➔ ✤❛ t❤ù❝ ❦❤æ♥❣✳ ❉♦ R [x1 , x2 , x3 ] ❧➔ tỗ t i s qi (x1 , x2 , x3 ) := αi x1 + βi x2 + γi x3 ❧➔ ✤❛ t❤ù❝ ❦❤æ♥❣✳ ❍❛② (αi , βi , γi ) = (0, 0, 0)✳ ❙✉② r❛ P ei ∈ (ker A ∩ ker B ∩ ker C)✱ tr♦♥❣ ✤â ei ❧➔ ❝ët tå❛ ✤ë ❝õ❛ ✈❡❝ tì ✤ì♥ ✈à t❤ù i tr♦♥❣ Rn ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t = k = dim (ker A ∩ ker B ∩ ker C)✳ ❱➟② A, B, C ❦❤æ♥❣ ❙❉❈✳ (ii) ●✐↔ sû det A = 0✳ (⇒) ●✐↔ sû A, B, C ❧➔ ❙❉❈✳ õ tỗ t tr P Cn×n s❛♦ ❝❤♦ P H AP = DA ✱ P H BP = DB ✈➔ P H CP = DC tr ữớ ú ỵ r DA , DB ✈➔ DC ❧➔ ❝→❝ ♠❛ tr➟♥ t❤ü❝✳ ❙✉② r❛ P −1 A−1 BP = P −1 A−1 P −H P H BP = DA −1 DB P −1 A−1 CP = P −1 A−1 P −H P H CP = DA −1 DC ✈➔ ❧➔ ❝→❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦✳ ❉♦ ✤â ♠é✐ A−1 B, A−1 C ❧➔ ❙❉❙ ữủt tữỡ ữỡ ợ tr ữớ t❤ü❝ DA −1 DB , DA −1 DC ✳ ❍ì♥ ♥ú❛✱ ❤➺ A−1 B, A−1 C ❧➔ ❙❉❙ tr♦♥❣ Rn×n () sỷ tỗ t tr Q s❛♦ ❝❤♦ Q−1 A−1 BQ = D1 ∈ Rn×n Q1 A1 CQ = D2 Rnìn ữớ ❝❤➨♦✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ ✺✷ q✉→t✱ ❣✐↔ sû D1 = diag (α1 In1 , , αk Ink ) ✈ỵ✐ αi = αj ∈ R, ∀i = j ✈➔ n1 + + nk = n✳ ❑❤✐ ✤â D1 = Q−1 A−1 BQ = Q−1 A−1 Q−H QH BQ = QH AQ −1 QH BQ −1 QH CQ ❙✉② r❛ QH AQ D1 = QH BQ = QH BQ = H H QH AQ D1 = D1 H QH AQ H = D1 QH AQ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✱ t❛ ❝â QH AQ = diag (A1 , , Ak ) ợ Ak Cni ìni rt i = 1, k ✳ ▼➦t ❦❤→❝✱ D2 = Q−1 A−1 CQ = Q−1 A−1 Q−H QH CQ = QH AQ ❙✉② r❛ QH AQ D2 = QH CQ = QH CQ = T QH AQ D2 = D2 H QH AQ = D2 QH AQ H H ✺✸ ❉♦ ✤â t❤❡♦ ▼➺♥❤ ✤➲ 1.1 t❛ ❝â D2 = diag (β1 In1 , , βk Ink ) ✈ỵ✐ βi = βj , ∀i = j ✈➔ n1 + + nk = n✳ ❙✉② r❛   α1 In1  A1   ✳✳✳  QH BQ =      αk Ink  ✳✳✳ Ak      = diag (α1 A1 , , αk Ak ) , ✈➔  β1 In1  ✳✳✳ QH CQ =    βk Ink    A1          ✳✳✳ Ak = diag (β1 A1 , , βk Ak ) Ai = Ai H tỗ t Qi ∈ Cni ×ni s❛♦ ❝❤♦    Qi H Qi = Ini   Ai = Qi H Λi Qi , tr õ i ữớ ợ i = 1, k ❑❤✐ ✤â  H Q1 Λ1 Q1  ✳✳✳ QH AQ =     H Q1  ✳✳✳ =   Qk H       Q k H Λk Q k   Λ1  ✳✳✳    Λk   Q1  ✳✳✳     Qk      ✺✹  Λ1  ˜ =Q    ✳✳✳ Λk   H ˜ , Q   tr♦♥❣ ✤â  H Q1  ✳✳✳ ˜= Q    Qk H      ❚÷ì♥❣ tü✱  α1 Λ1  ˜ QH BQ = Q    β1 Λ1  ˜ QH CQ = Q    ✳✳✳ αk Λk   H ˜ , Q    ✳✳✳ βk Λk   H ˜ Q   ❙✉② r❛ ˜ H QH AQQ ˜ = diag (Λ1 , , Λk ) , Q ˜ H QH BQQ ˜ = diag (α1 Λ1 , , αk Λk ) , Q ˜ H QH CQQ ˜ = diag (β1 Λ1 , , βk Λk ) Q ˜✳ ❱➟② A B C ữủ ợ tr P = QQ (2) ●✐↔ sû (ker A ∩ ker B ∩ ker C) = k > ❉♦ Rn ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❊✉❝❧✐❞ ♥➯♥ t❛ ❧➜② {u1 , , uk , uk+1 , , un } ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ❝õ❛ (ker A ∩ ker B ∩ ker C)✳ ✣➦t U = ([u1 ]e , , [un ]e ) = (u1 , , un ) ❧➔ ✺✺ ♠❛ tr➟♥ ✤ê✐ ❝ì sð tø (e) s❛♥❣ (u)✳ ❑❤✐ ✤â U H U = In ✳ ❍ì♥ ♥ú❛✱   H u1    uH   2 H U AU =   A u1 , u2 , , un  ✳✳✳      uH n = uH , ∀i, j = 1, n i Auj   0k , = A˜  H  u1    uH   2 H U BU =   B u1 , u2 , , un  ✳✳✳      uH n = uH , ∀i, j = 1, n i Buj   0k , = ˜ B ✈➔   H u1    uH   2 H U CU =   C u1 , u2 , , un  ✳✳✳      uH n = uH , ∀i, j = 1, n i Cuj ✺✻   0k  = C˜ ˜ B, C C(nk)ì(nk) rt ợ A, ker C > t tỗ t dim ker A˜ ∩ ker B ˜ ∩ ker C˜ ⊆ Cn−k = x˜ ∈ ker A˜ ∩ ker B   0k ❙✉② r❛ = x =   ∈ Cn×1 ✳ ❑❤✐ ✤â x˜  0k U H AU x =   0k U H BU x =   0k U H CU x =  0   x = 0, A˜   x = 0, ˜ B   x = C˜ ❙✉② r❛ = y = U x ∈ ker A, = y = U x ∈ ker B, = y = U x ∈ ker C ✺✼ ❉♦ ✤â y ∈ (ker A ∩ ker B ∩ ker C)✳ ❍ì♥ ♥ú❛✱ ∀j = 1, k, t❛ ❝â H uH j U x = uj y = uH j u1 , , uk , uk+1 , , un x T H H = uH x j u1 , , uj uk , uj uk+1 , , uj un   0 ✳  ✳✳  j   ∨ = 0, , 0, , , 0, ,   0     x˜ = ❙✉② r❛ y ⊥ (ker A ∩ ker B ∩ ker C)✳ ❚ø ✤â s✉② r❛ y = 0✱ ♠➙✉ t❤✉➝♥✳ ❉♦ ˜ ∩ ker C˜ = 0✳ ✤â dim ker A˜ ∩ ker B ˜ B, ˜ C˜ ❧➔ SDC ✳ ❚❤➟t ✈➙②✱ ❍ì♥ ♥ú❛✱ A, B, C ❧➔ SDC ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A, ⇔ ⇔ ˜ B, ˜ C˜ ❧➔ ❙❉❈ A,    ♠é✐ A˜−1 B, A˜−1 C ❧➔ ❙❉❙ tr♦♥❣ Rn×n    ❤➺ A˜−1 B, A˜−1 C ❧➔ SDS tr♦♥❣ Rn×n  −1    −1       0k  0k  0k  0k        ✈➔     ❧➔ ❙❉❙             ˜ ˜ ˜ ˜  B A C A        −1    −1         0k  0k  0k  0k      ,    ❧➔ ❙❉❙  ❤➺              ˜ A˜ B A˜ C˜ ✺✽ ⇔ ⇔ ⇔     P H AP −1 P H BP , P H AP −1 P H CP  −1   ❤➺ P H AP P H BP , P H AP    P −1 A−1 BP, P −1 A−1 CP ❧➔ ❙❉❙ −1 ❧➔ ❙❉❙ P H CP ❧➔ ❙❉❙    ❤➺ P −1 A−1 BP, P −1 A−1 CP ❧➔ ❙❉❙    ♠é✐ A−1 B, A−1 C ❧➔ ❙❉❙ tr♦♥❣ Rn×n    ❤➺ A−1 B, A−1 C ❧➔ SDS tr♦♥❣ Rn×n ⇔ A, B, C ❧➔ ❙❉❈ ▼➺♥❤ ✤➲ ✸✳✻✳ ❍➺ ❜❛ ♠❛ tr➟♥ ❍❡r♠✐t A, B, C ∈ Hn SDC tỗ t ởt ♠❛ tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣ X t❤ä❛ ♠➣♥    X 0,   AXB = BXA, BXC = CXB, CXA = AXC ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✷✱ {A, B, C} tỗ t↕✐ P ∈ Cn×n ❦❤↔ ♥❣❤à❝❤ s❛♦ ❝❤♦ P H AP, P H BP ✈➔ P H CP ✤æ✐ ♠ët ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ●å✐ P = QU, U H U = I, QH = Q ✭✯✯✮ 0, ❧➔ ♠ët ♣❤➙♥ t➼❝❤ P♦❧❛r ❝õ❛ P ✳ ❑❤✐ ✤â P H AP = U H QH AQU = U H (QAQ) U, ✈➔ t÷ì♥❣ tü ❝❤♦ ❝→❝ ♠❛ tr➟♥ P H BP ✱ P H CP ✳ ❚ø ✤â ❤➺ ✭ ✯✯ ✮ ✤æ✐ ♠ët ❣✐❛♦ ❤♦→♥ ♥❤❛✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ QAQ, QBQ, QCQ ✺✾ ✤æ✐ ♠ët ❣✐❛♦ ❤♦→♥ ♥❤❛✉✳ ✣➦t X = Q2 = QH Q✳ ❑❤✐ ✤â     AXB = BXA,       (QAQ)(QBQ) = (QBQ)(QAQ),       AXC = CXA,  (QAQ)(QCQ) = (QCQ)(QAQ), ⇔       BXC = CXB,      (QBQ)(QCQ) = (QCQ)(QCQ)    Rn×n X ✻✵ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉✿ (1) ❚r➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➺ ❤❛✐ ❤♦➦❝ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ õ tữỡ ỗ tớ ữủ t a) ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➺ ❤❛✐ tr ố ự õ tữỡ ỗ t❤í✐ ✤÷đ❝ t❤ỉ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ❝❤➨♦ ❤â❛ t÷ì♥❣ ✤÷ì♥❣ ỗ ỗ tớ ữủ ✷✳✷✮✳ b) ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➺ ❤❛✐ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❧➔ ❝❤➨♦ ❤â❛ t÷ì♥❣ ✤➥♥❣ ỗ tớ ữủ tổ q từ tử ỷ ✭①❡♠ ▼➺♥❤ ✤➲ ✷✳✸✮✳ c) ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➺ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❧➔ ❝❤➨♦ õ tữỡ ỗ tớ ữủ tổ q õ tữỡ ữỡ ỗ ỗ tớ ữủ ▼➺♥❤ ✤➲ ✷✳✹ ✈➔ ✷✳✻✮✳ d) ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➺ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❧➔ õ tữỡ ỗ tớ ữủ tổ q từ tö❝ ♥û❛ ①→❝ ✤à♥❤ ✭①❡♠ ▼➺♥❤ ✤➲ ✷✳✺✮✳ (2) ❚r➻♥❤ ❜➔② ❦➳t q✉↔ t÷ì♥❣ tü ♥❤÷ tr➯♥ ✤➙② ❝❤♦ ❤➺ ❤❛✐ ✭①❡♠ ❝→❝ ▼➺♥❤ ✤➲ ✸✳✶✱ ✸✳✷ ✈➔ ✸✳✸✮ ✈➔ ❜❛ ✭①❡♠ ❝→❝ ▼➺♥❤ ✤➲ ✸✳✹✱ ✸✳✺ ✈➔ ✸✳✻✮ ♠❛ tr➟♥ ❍❡r♠✐t✳ ✻✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❘✳❆ ❍♦r♥ ❛♥❞ ❈✳❘ ❏♦❤♥s♦♥✱ ▼❛tr✐① ❛♥❛❧②sts✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✽✺✳ ❬✷❪ ❘✳ ❏✐❛♥❣ ❛♥❞ ❉✳ ▲✐✱ ❙✐♠✉❧t❛♥❡♦✉s ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ♦❢ ♠❛tr✐❝❡s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s ✐♥ q✉❛❞r❛t✐❝❛❧❧② ❝♦♥str❛✐♥❡❞ q✉❛❞r❛t✐❝ ♣r♦❣r❛♠♠✐♥❣✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ✷✻✭✸✮✿ ✶✻✹✾ ✕ ✶✻✻✽✱ ✷✵✶✻✳ ❬✸❪ ❚✳❍✳▲❡ ❛♥❞ ❚✳◆✳◆❣✉②❡♥✱ ❊q✉✐✈❛❧❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r s✐♠✉❧t❛♥❡♦✉s ❞✐✲ ❛❣♦♥❛❧✐③❛t✐♦♥ ✈✐❛ ✯✲❝♦♥❣r✉❡♥❝❡ ♦❢ ❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡✱ ♣r❡♣r✐♥t✱ ✷✵✷✵✳ ❆✈❛✐❧❛❜❧❡ ❛t✿ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✷✵✵✼✳✶✹✵✸✹✳ ... ✈➜♥ ✤➲ ♥➯✉ tr➯♥ ✤è✐ ✈ỵ✐ ❤➺ ❤❛✐ ❤♦➦❝ ❜❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝✱ tr rt õ tữỡ ỗ tớ tr tr trữớ số ỗ ữỡ ❝❤➼♥❤✳ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët ▲✉➟♥ ✈➠♥ ✧ sè ❦✐➳♥... ▼➺♥❤ ✤➲ ✶✳✶✳ ◆➳✉ A = diag (α1In , , αk In ) ✈ỵ✐ αi = αj ∈ F ∀i = 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♥❣