Một số mô hình phân tích thành phần chính ba chiều

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ◆●❯❨➍◆ ❚❍➚ ⑩■ ▼❨ ▼❐❚ ❙➮ ▼➷ ❍➐◆❍ P❍❹◆ ❚➑❈❍ ❚❍⑨◆❍ P❍❺◆ ❈❍➑◆❍ ❇❆ ❈❍■➋❯ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ◆●❯❨➍◆ ❚❍➚ ⑩■ ▼❨ ▼❐❚ ❙➮ ▼➷ ❍➐◆❍ P❍❹◆ ❚➑❈❍ ❚❍⑨◆❍ P❍❺◆ ❈❍➑◆❍ ❇❆ ❈❍■➋❯ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✷ ữớ ữợ ▲❹▼ ❚❍➚ ❚❍❆◆❍ ❚❹▼ ❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ✈➲ ✤➲ t➔✐ ✏▼ët sè ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ❜❛ t tổ ữợ sỹ ữợ ❝õ❛ ❚❙✳ ▲➙♠ ❚❤à ❚❤❛♥❤ ❚➙♠ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝æ♥❣ ❜è ✤➸ ❜↔♦ ✈➺ ♠ët ❤å❝ ✈à ♥➔♦ ❝❤♦ tỵ✐ t❤í✐ ✤✐➸♠ ♥➔②✳ ❈→❝ ♥ë✐ ❞✉♥❣ ✈➔ ❦➳t q✉↔ sû ❞ö♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➲✉ ❝â tr➼❝❤ ❞➝♥ ✈➔ ú t ỗ ố õ t ý ❣✐❛♥ ❧➟♥✱ tæ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❇➻♥❤ ✣à♥❤✱ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✷✵ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ◆❣✉②➵♥ ❚❤à ⑩✐ ▼② ✐ ▼ö❝ ❧ö❝ ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲❮■ ◆➶■ ✣❺❯ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✸ ✶✳✶ ▼❛ tr➟♥ ✸ ✶✳✷ ❱➨❝tì r✐➯♥❣✳ ●✐→ trà r✐➯♥❣✳ tr õ ữợ ✈➨❝tì✳ ❚➼❝❤ ❑r♦♥❡❝❦❡r ✈➔ t➼❝❤ ❑❛tr✐✲❘❛♦ ❝õ❛ ❝→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♠❛ tr➟♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✐✐✐ ✻ ✽ ▼❐❚ ❙➮ ▼➷ ❍➐◆❍ P❍❹◆ ❚➑❈❍ ❍❆■ ❈❍■➋❯ ❱⑨ Ù◆● ❉Ö◆● ✶✵ ✷✳✶ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ P❤➙♥ t➼❝❤ ❣✐→ trà ❦➻ ❞à ✭❙❱❉✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷✳✶ P❤➙♥ t➼❝❤ ❣✐→ trà ❦➻ ❞à ✶✶ ✷✳✷✳✷ ❚❤✉➟t t♦→♥ t➻♠ ❙❱❉ ❝õ❛ ♠ët ♠❛ tr➟♥ ✷✳✷✳✸ ❱➼ ❞ö ✷✳✷✳✹ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠❛ tr➟♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❙❱❉ ❝õ❛ ♥â ✷✳✸ ✷✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✶✺ P❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ✭P❈❆✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ị tữ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✷ P❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✸ ❚➻♠ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ❝õ❛ ❜➔✐ t♦→♥ P❈❆ t❤æ♥❣ q✉❛ ❙❱❉ ✳ ✶✾ ✷✳✸✳✹ ❚➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ P❈❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✺ ❚❤✉➟t t♦→♥ t➻♠ P❈❆ ❝õ❛ ♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ì ữủ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❙❱❉ ✈➔ P❈❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✐✐ ✸ ✷✳✹✳✶ Ù♥❣ ❞ö♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ①➜♣ ①➾ ❤↕♥❣ t❤➜♣ tèt ♥❤➜t ❝õ❛ ♠❛ tr➟♥ ✷✷ ✷✳✹✳✷ Ù♥❣ ❞ö♥❣ tr♦♥❣ ①û ❧➼ ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✸ Ù♥❣ ❞ö♥❣ tr♦♥❣ ❊✐❣❡♥❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ▼❐❚ ❙➮ ▼➷ ❍➐◆❍ P❍❹◆ ❚➑❈❍ ❚❍⑨◆❍ P❍❺◆ ❈❍➑◆❍ ❇❆ ❈❍■➋❯ ✸✷ ✸✳✶ ▼↔♥❣ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✶✳✶ ❇❛ ❧♦↕✐ ✈➨❝tì ❝õ❛ ♠↔♥❣ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✶✳✷ ❇❛ ❧♦↕✐ ❧→t ❝➢t ❝õ❛ ♠↔♥❣ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✶✳✸ ❍↕♥❣ ❝õ❛ ♠↔♥❣ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ✸✳✸ ✸✳✹ ▼æ ❤➻♥❤ ❈❛♥❞❡❝♦♠♣✴P❛r❛❢❛❝ ✭❈P✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✶ ▼æ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✷ ❚❤✉➟t t♦→♥ t➻♠ ♥❣❤✐➺♠ ❈P ❝õ❛ ♠ët ♠↔♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✷✳✸ ❱➼ ❞ö ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼æ ❤➻♥❤ ❚✉❝❦❡r✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸✳✶ ▼æ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸✳✷ ❚❤✉➟t t♦→♥ ✹✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❈P ✈➔ ❚✉❝❦❡r✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ❑➌❚ ▲❯❾◆ ✹✾ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✺✵ ◗✉②➳t ✤à♥❤ ❣✐❛♦ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ✐✐✐ ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✷✳✶ P❤➙♥ t➼❝❤ ❤❛✐ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❱➼ ❞ư ✈➲ ♣❤÷ì♥❣ s❛✐ ❝õ❛ ❞ú ❧✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉✳ ✭❛✮ ❈❤✐➲✉ ✶✶ tự õ ữỡ s t ợ rở ❝õ❛ ✤÷í♥❣ ❤➻♥❤ ❝❤✉ỉ♥❣✮ ♥❤ä ❤ì♥ ❝❤✐➲✉ t❤ù ♥❤➜t✳ ✭❜✮ ❈↔ ❤❛✐ ❝❤✐➲✉ ❝â ♣❤÷ì♥❣ s❛✐ ✤→♥❣ ❦➸✳ P❤÷ì♥❣ s❛✐ ❝õ❛ ♠é✐ ❝❤✐➲✉ ❧➔ ♣❤÷ì♥❣ s❛✐ ❝õ❛ t❤➔♥❤ ♣❤➛♥ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❧➜② tr➯♥ t♦➔♥ ❜ë ❞ú ❧✐➺✉✳ P❤÷ì♥❣ s❛✐ t➾ ❧➺ t❤✉➟♥ ✈ỵ✐ ✤ë ♣❤➙♥ t→♥ ❝õ❛ ❞ú ❧✐➺✉✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸ ❷♥❤ ❣è❝ ✷✼ ✷✳✹ ❷♥❤ ❤✐➺✉ ❝❤➾♥❤ ✈ỵ✐ k = 10 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❷♥❤ ❤✐➺✉ ❝❤➾♥❤ ✈ỵ✐ k = 20 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✻ ❷♥❤ ❤✐➺✉ ❝❤➾♥❤ ✈ỵ✐ k = 50 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✼ ❷♥❤ ❤✐➺✉ ❝❤➾♥❤ ✈ỵ✐ k = 100 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✽ ❱➼ ❞ö ✈➲ ↔♥❤ ❝õ❛ ♠ët ♥❣÷í✐ tr♦♥❣ ❨❛❧❡ ❋❛❝❡ ❉❛t❛❜❛s❡ ✷✳✾ ❈→❝ ❡✐❣❡♥❢❛❝❡s t➻♠ ✤÷đ❝ ❜➡♥❣ P❈❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ tr ố ữợ ữủ s r tứ s ữợ õ ♥❤✐➲✉ ♥❤✐➵✉ ♥❤÷♥❣ ✈➝♥ ♠❛♥❣ ♥❤ú♥❣ ✤➦❝ ✤✐➸♠ r✐➯♥❣ ♠➔ ♠➢t ♥❣÷í✐ ❝â t❤➸ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶ ▼↔♥❣ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❈→❝ ✈➨❝tì ❝ët✱ ❞á♥❣✱ ✈➔ ❧ỵ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ❈→❝ ❧→t ❝➢t ♥❣❛♥❣✱ ✤ù♥❣✱ ✈➔ ❝❤➼♥❤ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹ P❤➙♥ t➼❝❤ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✐✈ ▲❮■ ◆➶■ ✣❺❯ ❚r♦♥❣ t❤è♥❣ ❦➯✱ ♣❤➙♥ t➼❝❤ ❞ú ❧✐➺✉ ❧➔ ♠ët ❦❤➙✉ ✈ỉ ❝ị♥❣ q✉❛♥ trå♥❣ ✤➸ t➻♠ ❤✐➸✉ ❝→❝ t❤ỉ♥❣ t✐♥ ❝â tr♦♥❣ ❞ú ❧✐➺✉✳ ❈ị♥❣ ✈ỵ✐ sü ❜ị♥❣ ♥ê t❤ỉ♥❣ t✐♥✱ ❝→❝ ❞ú ❧✐➺✉ ♥❣➔② ❝➔♥❣ ❧ỵ♥ ✈➔ trð ♥➯♥ ✤❛ ❞↕♥❣ r➜t ♥❤✐➲✉✳ ❚ø ✤â ❝ơ♥❣ ❧➔♠ ①✉➜t ❤✐➺♥ ♥❣➔② ❝➔♥❣ ♥❤✐➲✉ ♠ỉ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ❞ú ❧✐➺✉ ♥❤➡♠ ✤→♣ ù♥❣ ♥❤✉ ❝➛✉ t➻♠ ❤✐➸✉ t❤ỉ♥❣ t✐♥ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝✳ ▼ư❝ t✐➯✉ ❝õ❛ ❝→❝ ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ❧➔ t➻♠ ❤✐➸✉ ❝→❝❤ ❜✐➸✉ ❞✐➵♥ ❞ú ❧✐➺✉ s❛♦ ❝❤♦ ✤ì♥ ❣✐↔♥ ♠➔ ❧÷đ♥❣ t❤ỉ♥❣ t✐♥ ❜à ♠➜t ✤✐ ❧➔ ➼t ♥❤➜t✳ ▲✉➟♥ ✈➠♥ ♥➔② ♥❤➡♠ t➻♠ ❤✐➸✉ ✈➲ ❝→❝ ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ❞ú ❧✐➺✉✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ t➻♠ ❤✐➸✉ ✈➲ P❤➙♥ t➼❝❤ ❣✐→ trà ❦➻ ❞à ✭❙✐♥❣✉❧❛r ✈❛❧✉❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ❙❱❉✮ ✈➔ P❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ✭Pr✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✲ P❈❆✮ tr♦♥❣ ♠↔♥❣ ❤❛✐ ❝❤✐➲✉ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ❝❤ó♥❣✱ t➻♠ ❤✐➸✉ ✈➲ ♠ỉ ❤➻♥❤ ❈P ✈➔ ♠ỉ ❤➻♥❤ ❚✉❝❦❡r✸ tr♦♥❣ ♠↔♥❣ ❜❛ ❝❤✐➲✉✳ ◆❣♦➔✐ ♠ö❝ ❧ö❝✱ ❜↔♥❣ ❞❛♥❤ s→❝❤ ❤➻♥❤ ✈➩✱ ❧í✐ ♥â✐ ✤➛✉ ✈➔ ❦➳t ❧✉➟♥✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ởt số tự ỡ s ỵ tt ♠❛ tr➟♥✱ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧➼ ♣❤ê tr ố ự t õ ữợ ✈➨❝tì✱ t➼❝❤ ❑r♦♥❡❝❦❡r✱ ✈➔ t➼❝❤ ❑❤❛tr✐ ✲ ❘❛♦ ❝õ❛ ❝→❝ ♠❛ tr➟♥✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè ♠ỉ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ❤❛✐ ❝❤✐➲✉ ✈➔ ù♥❣ ❞ư♥❣✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ✈➲ ♣❤➙♥ t➼❝❤ ❤❛✐ ❝❤✐➲✉✱ ♣❤➙♥ t➼❝❤ ❣✐→ trà ❦➻ ❞à✱ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤✱ ✈➔ tr➻♥❤ ❜➔② ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ❤❛✐ ♠ỉ ❤➻♥❤ ♥➔②✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè ♠ỉ ❤➻♥❤ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ❜❛ ❝❤✐➲✉✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♠↔♥❣ ❜❛ ❝❤✐➲✉✱ ♠æ ❤➻♥❤ ❈❛♥❞❡✲ ❝♦♠♣✴P❛r❛❢❛❝ ✭❈P✮✱ ♠æ ❤➻♥❤ ❚✉❝❦❡r✸✱ ✈➔ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❤❛✐ ♠ỉ ❤➻♥❤ ♥➔②✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t ữợ sỹ ữợ t t ❚❤à ❚❤❛♥❤ ❚➙♠✳ ✶ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✈➲ sü ❞➝♥ ❞➢t✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❈æ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ◆❤➙♥ ✤➙②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t t t ợ ỵ ❚❤➛② ❈ỉ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥ ✈➲ sü t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱ ❝ơ♥❣ ♥❤÷ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❑❤â❛ ❧✉➟♥ ỗ tớ tổ ụ ỷ ỡ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❝❤ó♥❣ tỉ✐ ✈➲ ♠➦t t✐♥❤ t❤➛♥ tr♦♥❣ t❤í✐ ❣✐❛♥ q✉❛✳ ▼➦❝ ❞ò ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ tr♦♥❣ ✈✐➺❝ t➻♠ tá✐✱ ❤å❝ ❤ä✐ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝õ❛ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ tổ rt ữủ sỹ õ ỵ ỵ ổ õ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❇➻♥❤ ✣à♥❤✱ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ⑩✐ ▼② ✷ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ tr➼❝❤ ❞➝♥ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱✹✱✻✱✶✷❪✳ ✶✳✶ ▼❛ tr➟♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼❛ tr➟♥ ❝ï t❤➔♥❤ m ❞á♥❣ ✈➔ n ❝ët✳ ▼❛ tr➟♥ mìn A ù ởt ỗ mìn a a12  11   a21 a22 A=  ✳✳ ✳ ✳  ✳ ✳  am1 am2 mn sè t❤ü❝ ✤÷đ❝ s➢♣ ①➳♣ t❤÷í♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ♥❤÷ s❛✉✿ a1n    a2n  , ✳  ✳  ✳  amn ❤♦➦❝  a a12  11   a21 a22 A=  ✳✳ ✳ ✳  ✳ ✳  am1 am2 ❤♦➦❝ A = (aij )m×n tr♦♥❣ ✤â aij a1n    a2n  , ✳  ✳  ✳  amn ❧➔ ♣❤➛♥ tû ❝õ❛ ♠❛ tr➟♥ ♥➡♠ tr➯♥ ❞á♥❣ i✱ ❝ët j ✈ỵ✐ i = 1, m, j = 1, n✳ ❱ỵ✐ m = n✱ t❛ ❣å✐ ♠❛ tr➟♥ ❝ï a11 , a22 , , amm m×m ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ m✳ ❈→❝ ♣❤➛♥ tû ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❝õ❛ ♠❛ tr➟♥✳ ✸ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ▼❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❝â ♠å✐ ♣❤➛♥ tû ♥➡♠ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ ✶✱ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ❜➡♥❣ ✵✳ ❚❛ ❦➼ ❤✐➺✉ ♠❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n ❧➔ In ✈➔ ♥â ❝â ❞↕♥❣ ♥❤÷ s❛✉✿       0 0   In =  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳   0 ❚r♦♥❣ trữớ ủ ổ ú ỵ tr➟♥✱ t❛ ❦➼ ❤✐➺✉ ♠❛ tr➟♥ ✤ì♥ ✈à ❜ð✐ I✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ▼❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝â ❝→❝ ♣❤➛♥ tû ♥➡♠ ♥❣♦➔✐ D ❝â ❞↕♥❣   a   11    a22    D= ✳ ✳ ✳  ✳✳ ✳ ✳ ✳ ✳  ✳ ✳ ✳    0 ann ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ ✵✳ ▼❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ♥❤÷ s❛✉ diag (a11 , a22 , , ann ) ❚❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ tr ữớ ợ a11 , a22 , , ann ❧➔ ❝→❝ ♣❤➛♥ tû ♥➡♠ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤✳ ▼❛ tr➟♥ ❝❤➾ ❝â ♠ët ❞á♥❣ ✤÷đ❝ ❣å✐ ❧➔ ✈➨❝tì ❞á♥❣✳ ▼❛ tr➟♥ ❝❤➾ ❝â ♠ët ❝ët ✤÷đ❝ ❣å✐ ❧➔ ✈➨❝tì ❝ët✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ▼❛ tr➟♥ ✈✉æ♥❣ t↕✐ ♠❛ tr➟♥ ✈✉æ♥❣ B ❝➜♣ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♠❛ tr➟♥ n A✱ t❤ä❛ ♠➣♥ ❧➔ ♠❛ tr➟♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤è✐ ①ù♥❣ ♥➳✉ A ❝➜♣ n ữủ tr tỗ AB = BA = In ✳ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❈❤♦ ♠❛ tr➟♥ ▼❛ tr➟♥ ✈✉æ♥❣ A ❑❤✐ ✤â B ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ A−1 ✳ A = (aij )m×n ✳ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ A✱ ❦➼ ❤✐➺✉ AT ✱ AT = (aji )n×m ✳ ✤÷đ❝ ❣å✐ ❧➔ ✤è✐ ①ù♥❣ ♥➳✉ AT = A✱ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ♣❤↔♥ AT = −A✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ▼❛ tr➟♥ ✈✉ỉ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ trü❝ ❣✐❛♦ ♥➳✉ AT A = AAT = I✳ ✹ T ✈ỵ✐ = ai1 ai2 T , bi = bi1 bi2 T , c =i ci1 ci2 , i = 1, 2✳    a11 b11 c11 + a21 b21 c21 = −1       a11 b12 c11 + a21 b22 c21 =       a12 b11 c11 + a22 b21 c21 =      a1 b c + a2 b c = 2                     c1 = 0✱ tø   a2 b2 c2 = −1   1 a21 b22 c21 =     a2 b c = ◆➳✉ (1), (2), (3) 1 ❚ø 2 ❑❤✐ ✤â✱ t❛ ❝â (1) (2) (3) (4) a11 b11 c12 + a21 b21 c22 = (5) a11 b12 c12 + a21 b22 c22 = (6) a12 b11 c12 + a22 b21 c22 = (7) a12 b12 c12 + a22 b22 c22 = (8) s✉② r❛ = 0, b21 = 0, c21 = ⇒ a21 ⇒ b22 = ⇒ a22 = (6) ✈➔ b22 = s✉② r❛ a11 , b12 , c12 = 0✳ ▼➦t ❦❤→❝✱ tø (7) ✈➔ a22 = s✉② r❛ a12 , b11 , c12 = 0✳ ❉♦ ✤â a12 b12 c12 +a22 b22 c22 = 0✳ ❱➟② (8) ✈ỉ ♥❣❤✐➺♠✱ ❦➨♦ t❤❡♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈ỉ ♥❣❤✐➺♠✳ ❱➻ ✈➟②✱ ✤➸ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ t❤➻ ❚÷ì♥❣ tü✱ t❛ ❝â c11 = 0✳ c21 , b11 , b21 , a11 , a21 = 0✳ ◆➳✉ b11 =    =0 b22     a1 b1 c1 = 2  a11 b12 c12 =     a1 b1 c1 = 2 t❤➻ tø (2)✱ (4)✱ (6) ✈➔ (8) s✉② r❛ ❍➺ ✈æ ♥❣❤✐➺♠✳ ❉♦ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈ỉ ♥❣❤✐➺♠✳ ❱➟② ✤➸ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ t❤➻ b11 = 0✳ ❚÷ì♥❣ tü✱ b22 = 0✳ ❚ø ✤â s✉② r❛ (6) T ổ tỗ t = ai1 ai2 ổ T , bi = bi1 bi2 T , c =i ci1 ci2 X = a1 ◦ b ◦ c + a2 ◦ b ◦ c ✳ rữớ ủ sỷ X ữủ ữợ ❞↕♥❣ X = a1 ◦ b1 ◦ c1 + a2 ◦ b2 ◦ c2 , +a3 ◦ b3 ◦ c3 ✸✽ , i = 1, ✤➸ T ✈ỵ✐ = ai1 ai2 T , bi = bi1 bi2 T , c =i ci1 ci2 , i = 1, 2✳ ❑❤✐ ✤â✱ t❛ ❝â    a11 b11 c11 + a21 b21 c21 + a31 b31 c31 = −1       a11 b12 c11 + a21 b22 c21 + a31 b32 c31 =       a12 b11 c11 + a22 b21 c21 + a32 b31 c31 =      a1 b c + a2 b c + a3 b c = 2                    2 2 a11 b11 c12 + a21 b21 c22 + a31 b31 c32 = a11 b12 c12 + a21 b22 c22 + a31 b32 c32 = a12 b11 c12 + a22 b21 c22 + a32 b31 c32 = a12 b12 c12 + a22 b22 c22 + a32 b32 c32 = ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ t❛ ❝❤å♥ ✤÷đ❝             −1 −1 −1 a1 =   , b1 =   , c1 =   , a2 =   , b2 =   , c2 =   , −1 −1       −1 a3 =   , b3 =   , c3 =   1 ❱➟② rank(X) = 3✳ ✸✳✷ ▼æ ❤➻♥❤ ❈❛♥❞❡❝♦♠♣✴P❛r❛❢❛❝ ✭❈P✮ ✸✳✷✳✶ ▼æ ❤➻♥❤ ◆➠♠ ✶✾✼✵✱ ❈❛rr♦❧❧ ✈➔ ❈❤❛♥❣ ❬✺❪ ✈➔ ❍❛rs❤♠❛♥ ❬✼❪ ✤➣ ✤ë❝ ❧➟♣ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❞↕♥❣ ❝õ❛ P❈❆ ợ t tữỡ ự P t➼❝❤ ❝❤➼♥❤ t➢❝✮ ✈➔ P❛r❛❢❛❝ ✭P❤➙♥ t➼❝❤ ♥❤➙♥ tû s♦♥❣ s♦♥❣✮ ✭✈✐➳t ❣å♥ ❧➔ ♠æ ❤➻♥❤ ❈P✮✳ P❤➙♥ t➼❝❤ ❈P ❝õ❛ ♠ët ♠↔♥❣ ❜❛ ❝❤✐➲✉ ❧➔ ♣❤➙♥ t➼❝❤ ♠↔♥❣ ♥➔② t❤➔♥❤ tê♥❣ ❝õ❛ ❝→❝ ♠↔♥❣ ❤↕♥❣ ✶✳ ▼æ ❤➻♥❤ ❈P ❧➔ tê♥❣ ❝õ❛ ♠ët ♣❤➙♥ t➼❝❤ ❈P ✈➔ ♠↔♥❣ ❞÷✳ ❱➲ ♠➦t ❤➻♥❤ t❤ù❝✱ ❝❤♦ ❑❤✐ ✤â ♠æ ❤➻♥❤ ❈P ❝õ❛ X X ❧➔ ♠↔♥❣ ❝ï m × n × p✳ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉✿ q q Y (t) + E, gt (at ◦ bt ◦ ct ) + E = X= t=1 t=1 ✸✾ ✭✸✳✶✮ tr♦♥❣ ✤â E ❧➔ ♠↔♥❣ ❞÷✱ ✤ë ❞➔✐ ❜➡♥❣ ✶✱ gt q số trữợ trồ số t ♣❤➛♥ t❤ù ❤↕♥❣ ✶✱ ❣å✐ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥✳ ❈è ✤à♥❤ tè✐ t❤✐➸✉ E at ∈ Rm ✱ bt ∈ Rn ✱ ct ∈ Rp e2ijk ✳ = q✱ ❧➔ ❝→❝ ✈➨❝tì ❝â t✱ Y (t) = gt (at ◦ bt ◦ ct ) ❧➔ ❝→❝ ♠↔♥❣ ♣❤➙♥ t➼❝❤ ❈P ✭✸✳✶✮ ❝â ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ❣✐↔♠ ❑❤✐ ✤â ♠ỉ ❤➻♥❤ ❈P ✭✸✳✶✮ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ❜➡♥❣ ❤➻♥❤ ↔♥❤ ijk s❛✉✿ ❍➻♥❤ ✸✳✹✿ P❤➙♥ t➼❝❤ ❜❛ ❝❤✐➲✉ ❈❤ó♥❣ t❛ ❦➼ ❤✐➺✉ Rp×q ✱ A = [a1 aq ] ∈ Rm×q ✱ B = [b1 bq ] ∈ Rn×q ✈➔ C = [c1 cq ] ∈ ❣å✐ ❧➔ ❝→❝ ♠❛ tr➟♥ t❤➔♥❤ ♣❤➛♥✳ ❑❤✐ ✤â ❝❤ó♥❣ t❛ ❝â t❤➸ ✤÷❛ r❛ ❦➼ ❤✐➺✉ ❝❤♦ ♠ỉ ❤➻♥❤ ❈P ♥❤÷ s❛✉✳ ❚❛ ❝â X = g1 (a1 ◦ b1 ◦ c1 ) + + gq (aq ◦ bq ◦ cq ) + E = g1 c1 ∗ a1 bT1 ❑❤✐ ✤â✱ ✈ỵ✐ (12) Xk k = 1, , p, + · · · + gq cq ∗ aq bTq + E t❛ ❝â (12) = g1 ck1 a1 bT1 + + gq ckq aq bTq + Ek     a a  11   1q   ✳✳   ✳✳  = g1 ck1  ✳  b11 bn1 + + gq ckq  ✳  b1q bnq     am1 amq (12)    + Ek  a b a11 bn1 a b a1q bnq  11 11   1q 1q  ✳ ✳ ✳  ✳✳    ✳✳ ✳✳ ✳ ✳ ✳ = g1 ck1  ✳ ✳ ✳  + · · · + gq ckq  ✳  ✳ ✳     am1 b11 am1 bn1 amq b1q amq bnq (12) +Ek ✹✵   g c a b + + gq ckq a1q b1q g1 ck1 a11 bn1 + + gq ckq a1q bnq  k1 11 11     g1 ck1 a21 b11 + + gq ckq a2q b1q g1 ck1 a21 bn1 + + gq ckq a2q bnq    =   ✳ ✳ ✳✳ ✳ ✳ ✳   ✳ ✳   g1 ck1 am1 b11 + + gq ckq amq b1q g1 ck1 am1 bn1 + + gq ckq amq bnq (12) +Ek  a a12  11   a21 a22 =   ✳✳ ✳ ✳  ✳ ✳  am1 am2    a1q gc b b bn1   k1   11 21     a2q   g2 ck2  b12 b22 bn2     + E (12) k ✳  ✳ ✳ ✳  ✳ ✳ ✳  ✳✳ ✳✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  ✳ ✳ ✳  ✳ ✳ ✳     amq 0 gq ckq b1q b2q bnq (12) = ACk B T + Ek , ✈ỵ✐ ✭✸✳✷✮   gc   k1    g2 ck2  ✳ Ck =   ✳✳ ✳ ✳  ✳✳ ✳ ✳  ✳  ✳ ✳ ✳   0 gq ckq ❚÷ì♥❣ tü✱ t❛ ❝â (23) = BAi C T + Ei (13) = CBj AT + Ej Xi Xj (23) , i = 1, , m, (13) ✭✸✳✸✮ , j = 1, , n, ✭✸✳✹✮     ga gb   i1   j1       g2 ai2  g2 bj2  ✱ Bj =  ✳ ❉♦ ✤â✱ ❦❤æ♥❣ ♠➜t ✈ỵ✐ Ai =   ✳✳  ✳✳ ✳ ✳  ✳ ✳  ✳✳ ✳✳ ✳ ✳  ✳ ✳  ✳ ✳  ✳  ✳ ✳ ✳ ✳ ✳     0 aq ciq 0 gq bjq t➼♥❤ tê♥❣ q✉→t✱ ❝❤ó♥❣ t❛ ❣✐↔ sû r➡♥❣ ❝→❝ trå♥❣ sè gt ✤÷đ❝ ❜ị ✈➔♦ A ❤♦➦❝ B ❤♦➦❝ C✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ t❛ ❝â ♠ët ❝→❝❤ ❦❤→❝ ✤➸ ✈✐➳t ♠ỉ ❤➻♥❤ ❈P ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ t➼❝❤ ❑❤❛tr✐ ✲ ữ tr ữợ X(npìm) = (12) X1 = (C Xp(12) T   =  B) AT + E(np×m) , ✹✶ BC1 ✳ ✳ ✳ BCp    T  A + E(np×m)  ✭✸✳✺✮ gt tr♦♥❣ ✤â ✤÷đ❝ ❜ị ✈➔♦ C❀  (23) X1 X(mpìn) = gt ữủ ũ (23) Xm   =  CA1 ✳ ✳ ✳ CAm (13) X(mnìp) = X1 ữủ ũ T Xn(13)   =  (2.1) ✈➔ ❍➻♥❤ ✳ ✳ ✳ ABn   T  C + E(mn×p)  ✭✸✳✼✮ B ó r õ ởt sỹ tữỡ ỗ ỳ ổ ❤➻♥❤ ❍➻♥❤ AB1  A) C T + E(mn×p) , = (B gt ✭✸✳✻✮ A❀  tr♦♥❣ ✤â   T  B + E(mp×n)  C) B T + E(mp×n) , = (A tr♦♥❣ ✤â T  (3.4)✳ ❇ë ❜❛ (A, B, C) ❝õ❛ ♠æ ❤➻♥❤ ❈P ❤♦➦❝ ①➜♣ ①➾ ❤↕♥❣ q ❝õ❛ (2.1) ✈➔ ♠ỉ ❤➻♥❤ (3.1)✱ ❝ơ♥❣ ♥❤÷ ❣✐ú❛ t❤ä❛ ♠➣♥ ♠ỉ ❤➻♥❤ ❈P ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ X✳ ❈→❝ ♠❛ tr➟♥ A✱ B ✱ C ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♠❛ tr➟♥ t❤➔♥❤ ♣❤➛♥✳ ◆➳✉ ✈➔ E q = rank(X) t❤➻ (A, B, C) ❧➔ ♥❣❤✐➺♠ ✤ó♥❣ tèt ♥❤➜t ❝õ❛ ❈P✳ ◆➳✉ q < rank(X) ♥❤ä ♥❤➜t t❤➻ ♥❤➜t ❝õ❛ X✳ (A, B, C) ❉♦ ✤â✱ ❤↕♥❣ ❝õ❛ ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❈P tè✐ ÷✉ ❤♦➦❝ ①➜♣ ①➾ ❤↕♥❣ X ❧➔ sè q ♥❤ä ♥❤➜t s❛♦ ❝❤♦ X q tèt ❝â ♣❤➙♥ t➼❝❤ ❈P t♦➔♥ ♣❤➛♥✳ ✸✳✷✳✷ ❚❤✉➟t t♦→♥ t➻♠ ♥❣❤✐➺♠ ❈P ❝õ❛ ♠ët ♠↔♥❣ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû ❝→❝ trå♥❣ sè tr♦♥❣ ♠æ ❤➻♥❤ ✈➔♦ tr♦♥❣ ♠❛ tr➟♥ C✳ (3.1) ✤➲✉ ✤÷đ❝ ❜ị ❚❤✉➟t t♦→♥ ✤ì♥ ❣✐↔♥ ♥❤➜t ♣❤ị ❤đ♣ ợ ổ P ữủ tt t ♣❤÷ì♥❣ tè✐ t❤✐➸✉ ❧✉➙♥ ♣❤✐➯♥ ✭❆▲❙✮✱ ❝â ❣✐↔✐ t❤✉➟t ♥❤÷ s ã ữợ t ợ tr A✱ B ✱ C ❜➜t ❦➻✱ t❛ ❝â✿ p (12) εo = l (A, B, C) = Xk k=1 ✹✷ ACk B T ã ữợ ợ B ✈➔ C ❝è ✤à♥❤✱ t➻♠ A tèt ♥❤➜t t❤❡♦ ❝æ♥❣ t❤ù❝ s❛✉✿ −1 p T p (12) T A = Ck B T X k Ck B BCk k=1 • ữợ ợ A C B tốt t t ❝æ♥❣ t❤ù❝ s❛✉✿ −1 p p (12) T B = Ck AT Xk Ck A ACk k=1 ã ữợ ợ A ck = (A õ ã B k=1 ❝è ✤à♥❤✱ t➻♠ T T C = c1 c2 cp k=1 ❝è ✤à♥❤✱ t➻♠ B)T (A T C tèt ♥❤➜t t❤❡♦ ❝æ♥❣ t❤ù❝ s❛✉✿ −1 B) (12)) B)T V ec Xk (A , k = 1, , p ữợ ữợ o < ợ >0 trữợ õ ✸✳✷✳✸ ❱➼ ❞ư ❈❤♦ ❱ỵ✐ X ❧➔ ♠↔♥❣ ❝ï q = 3 ì ì ợ t   0     (12) X1 = 0 0 ,   0 ❝➢t ❝❤➼♥❤ ❞✐➺♥ (12) X2   −1     = 2    −4 ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ♣❤➛♥ t ợ = 107 t ữủ ♥❣❤✐➺♠ ❈P s❛✉   −0.4738     A = −0.6649 −0.6535 0.6599  ,   −0.5671 −0.5999 0.5835 0.4864 0.4616   0.7236 −0.6933 0.7086     B = 0.6068 −0.6507 0.6290 ,   0.3291 −0.3098 0.3203   0.2893 −0.2690 0.2790 , C= 0.9572 −0.9631 0.9603 ✈➔ ❝→❝ trå♥❣ sè t÷ì♥❣ ù♥❣ ❧➔ g1 = 2258.674✱ g2 = 2324.58✱ g3 = 4579.80✳ ✹✸ ✸✳✸ ▼æ ❤➻♥❤ ❚✉❝❦❡r✸ ✸✳✸✳✶ ▼æ ❤➻♥❤ ◆➠♠ ✶✾✻✻✱ ❚✉❝❦❡r ✤÷❛ r❛ ♠ỉ ❤➻♥❤ ❚✉❝❦❡r✸✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ q r s gtuv (at ◦ bu ◦ cv ) + E X= ✭✸✳✽✮ t=1 u=1 v=1 ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② r➡♥❣ ♠æ ❤➻♥❤ ❈P ♥â✐ tr➯♥✳ ❚❤➟t ✈➟②✱ ợ q =r =s (3.1) trữớ ủ t ❝õ❛ ♠æ ❤➻♥❤ ❚✉❝❦❡r✸ ✈➔ gtuv = ♥➳✉ (3.8) trð t❤➔♥❤ ♠ỉ ❤➻♥❤ (3.1)✳ ▼↔♥❣ G ∈ Rq×r×s ♥❤➙♥✳ ❚❤ỉ♥❣ t❤÷í♥❣✱ ❝→❝ ♠❛ tr➟♥ t❤➔♥❤ ♣❤➛♥ (t, u, v) = (t, t, t) ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ gtuv t❤➻ ♠ỉ ữủ A Rmìq B Rnìr C Rpìs ợ trð t❤➔♥❤ ✤ỉ✐ ♠ët trü❝ ❝❤✉➞♥ ✈ỵ✐ ♥❤❛✉✳ ▼ỉ ❤➻♥❤ ❚✉❝❦❡r✸ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ❝ỉ♥❣ t❤ù❝ s❛✉✿ ❚ø (3.8)✱ t❛ ❝â q r s gtuv cv ∗ at bTu X= + E t=1 u=1 v=1 ❙✉② r❛ q (12) Xk r s gtuv ckv at bTu = (12) + Ek t=1 u=1 v=1  a1t       a2t  (12)  = gtuv ckv   ✳✳  b1u b2u bnu + Ek  ✳  t=1 u=1 v=1   amt   a b a1t b2u a1t bnu  1t 1u    q s r  a2t b1u a2t b2u a2t bnu   + E (12) = ckv gtuv  k  ✳✳  ✳ ✳ ✳ ✳ ✳✳ ✳  ✳  ✳ ✳ v=1 t=1 u=1   amt b1u amt b2u amt bnu   r r a1t gtuv b1u a1t gtuv bnu  q  u=1 u=1 s   ✳ ✳ ✳✳   + E (12) ✳ ✳ = ckv ✳ ✳ ✳ k    r r v=1 t=1  amt gtuv b1u amt gtuv bnu q r s u=1 u=1 ✹✹  = = = =  a  1t   q  s r  a2t  r   g b gtuv bnu + Ek(12) ckv tuv 1u  ✳✳  u=1 u=1 t=1  ✳  v=1   amt     b b bn1 a   11 21  1t      q s b12 b22 bn2   a2t   + E (12)    g ckv k  ✳✳  t1v gt2v gtru  ✳✳ ✳ ✳  ✳ ✳ ✳✳ ✳   ✳ ✳ ✳ t=1  ✳  v=1     b1r b2r bnr amt  q  q q a g a g a1t gtrv  t=1 1t t1v t=1 1t t2v  t=1  q  q q   s a2t gt2v a2t gtrv   a2t gt1v  B T + E (12) t=1 t=1 ckv  k  t=1 ✳  ✳ ✳ ✳✳ ✳ ✳ ✳   v=1 ✳ ✳ ✳ ✳    q  q q amt gt1v amt gt2v amt gtrv t=1 t=1   t=1 g g g1rv a a12 a1q    11v 12v  11     s  a21 a22 a2q  g21v g22v g2rv  T  B + E (12)  ckv  k  ✳✳ ✳ ✳  ✳ ✳  ✳ ✳ ✳ ✳ ✳✳ ✳  ✳ ✳✳ ✳  ✳  ✳ ✳ ✳ ✳ ✳ ✳ v=1    gq1v gq2v gqrv am1 am2 amq s (12) ckv AGv B T + Ek = v=1 s = A ckv Gv (12) B T + Ek , v=1 tr♦♥❣ ✤â Gv ∈ Rq×r ❧➔ ❧→t ❝➢t ❝❤➼♥❤ ❞✐➺♥ t❤ù v ❝õ❛ G✳ ❚ø ✤â✱ ❝❤ó♥❣ t❛ õ t t ổ r ữợ tr ♥❤÷ s❛✉✿  T s T v=1 c1v BGv ✳ ✳ ✳      T =   A + Enp×m   sumsv=1 cpv BGTv    c B · · · c1s B GT  11  1 ✳  ✳  T  ✳ ✳✳ ✳  ✳ A + E =  ✳✳ ✳ ✳ ✳ np×m    cp1 B · · · cps B GTs Xnp×m = (12) X1 (12) Xp = (C ⊗ B) GT AT + Enp×m , ✹✺ ✭✸✳✾✮ tr♦♥❣ ✤â G = G1 · · · ✳ ❑❤✐ ✤â✱ t❛ ❝â t❤➸ ♠æ t↔ ♠é✐ ♣❤➛♥ tû ❝õ❛ Gs q r X ♥❤÷ s❛✉✿ s X = (A, B, C) G ⇔ xijk = ait bju ckv gtuv t=1 u=1 v=1 ◆➳✉ G ❧➔ ♠↔♥❣ ❝➜♣ q × q × q s✐➯✉ ❝❤➨♦ ✈ỵ✐ q ❧→t ❝➢t ❝❤➼♥❤     0 ··· g ···         0 · · ·   0 · · · 0   , G1 =   ✳✳ ✳✳ ✳ ✳ ✳✳  , · · · , Gq =  ✳✳ ✳✳ ✳ ✳ ✳  ✳  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳     0 ··· 0 · · · gq r=s=p ✈➔ t❤➻ ❦➼ ❤✐➺✉ ❝õ❛ ♠æ ❤➻♥❤ ❈P s (3.1) ữủ t ữợ X = (A, B, C) G + E ✳ ▼æ ❤➻♥❤ ❚✉❝❦❡r✸ ❝â t❤➸ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜ð✐ t❤✉➟t t♦→♥ ❆▲❙✳ ❱ỵ✐ ♠ỉ ❤➻♥❤ ❚✉❝❦❡r✸ s➩ ❝❤♦ ❝❤ó♥❣ t❛ ♠ët ①➜♣ ①➾ ❤↕♥❣ q ≤ m ✱ r ≤ n✱ s ≤ p✱ (q, r, s) ✤❛ t✉②➳♥ t➼♥❤ tèt ♥❤➜t ❝õ❛ X✳ ✸✳✸✳✷ ❚❤✉➟t t♦→♥ ✣➦t T X = Xm×np = Xnpìm tr õ Xnpìm ữủ ữ tr♦♥❣ (3.9)✳ ◆❤í ❑r♦♦♥❡♥❜❡r❣ ❛♥❞ ❉❡ ▲❡❡✉✇ ✭✶✾✽✵✮✱ ♠ỉ ❤➻♥❤ ❚✉❝❦❡r✸ trð ♥➯♥ ♣❤ị ❤đ♣ ❜➡♥❣ ❝→❝❤ ❣✐↔♠ t❤✐➸✉ ❤➔♠ s❛✉✿ X − AG (C ⊗ B)T A,B,C ✈ỵ✐ ❣✐↔ t❤✐➳t (3.9) A✱ B ✈➔ C , ✭✸✳✶✵✮ ❧➔ ❝→❝ ♠❛ tr➟♥ trü❝ ❣✐❛♦✳ ❚❤✉➟t t♦→♥ ❆▲❙ ❝❤♦ ổ r ữủ tỹ ữ s ã ữợ t ợ ã ữợ tr B C t A tr ỗ q ✈➨❝tì ❦➻ ❞à ✤➛✉ t✐➯♥ ❝õ❛ ♠❛ tr➟♥ B ❧➔ tr ỗ r tỡ t tr C tr ỗ s tỡ ❞à ✤➛✉ t✐➯♥ ❝õ❛ ♠❛ tr➟♥ Xm×nq (C ⊗ B)✳ ã ữợ tr Xnìmq (C A) ã ữợ tr Xpìmn (B A) G = AT X (C B) ã ữợ ã ữợ o ã ữợ ữợ ❝❤♦ ✤➳♥ ❦❤✐ = X − AG (C ⊗ B)T o < ợ trữợ õ ✸✳✹ ▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❈P ✈➔ ❚✉❝❦❡r✸ ▼ët tr♦♥❣ ♥❤ú♥❣ t➼♥❤ ♥➠♥❣ ❤➜♣ ❞➝♥ ♥❤➜t ❝õ❛ ❈P ❧➔ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♥â✳ ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♥❣❤✐➺♠ ❈P t❤÷í♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ♠ët ♠↔♥❣ ✤➣ ✤÷đ❝ ✤✐➲✉ ❝❤➾♥❤ ♣❤ị ❤đ♣ X = X − E✳ ❈❤ó♥❣ t❛ ❣✐↔ sû ❝→❝ trå♥❣ sè gt tr♦♥❣ t❤➔♥❤ ♣❤➛♥✳ ❈â t❤➸ t❤➜② r➡♥❣ ❝→❝ ♠❛ tr➟♥ t❤➔♥❤ ♣❤➛♥ (3.1) (A, B, C) tr♦♥❣ (3.1) ❝❤➾ ❝â t❤➸ ❧➔ ❞✉② ♥❤➜t t tữợ ởt ❝õ❛ (A, B, C) ❧➔ ♥❣❤✐➺♠ ❈P ❝õ❛ ♠❛ tr➟♥ ❤♦→♥ ✈à ✈➔ (3.1) Ta , Tb , Tc t❤➻ ✤÷đ❝ ❜ị ✈➔♦ ❝→❝ ♠❛ tr➟♥ A, B, C ✳ A, B, C = (AP Ta , BP Tb , CP Tc ) tr ữớ ợ ❧➔ ♥❤ú♥❣ ❧ü❛ ❝❤å♥ t❤❛② t❤➳ ❞✉② ♥❤➜t ❝â t❤➸ t❤➻ ♥❣❤✐➺♠ ❈P ❚❤➟t ✈➟②✱ ♥➳✉ tr♦♥❣ ✤â Ta Tb Tc = Iq ✳ (A, B, C) P ❧➔ ◆➳✉ ✤➙② ✤÷đ❝ ❣å✐ ❧➔ ❞✉② ♥❤➜t ✈➲ ❜↔♥ ❝❤➜t✳ ❱➼ ❞ư ✈ỵ✐ (A, B, C) q = 3✱ ✈➔ A = a1 a2 a3 ❧➔ ♥❣❤✐➺♠ ❈P t❤➻ ✤è✐ ✈ỵ✐ ♠❛ B = b1    tr➟♥ P = 0  ✱ Tb = diag {β1 , β2 , β3 }✱ Tc = diag {γ1 , γ2 , γ3 }✱ b2 b3 ✱ C = c1 c2 c3 ✳ ◆➳✉    1✱ Ta = diag {α1 , α2 , α3 }✱  0 t❛ ❝â A = AP Ta = α1 a3 α2 a1 α3 a2 = a1 a2 a3 , B = BP Tb = β1 b3 β2 b1 β3 b2 = b1 b2 b3 , C = CP Tc = γ1 c3 γ2 c1 γ3 c2 = c1 c2 c3 , ✈➔ aj ◦ bj ◦ cj = (α1 a3 ) ◦ (β1 b3 ) ◦ (γ1 c3 ) + (α2 a1 ) ◦ (β2 b1 ) ◦ (γ2 c1 ) j=1 + (α3 a2 ) ◦ (β3 b2 ) ◦ (γ3 c2 ) = α1 β1 γ1 (a3 ◦ b3 ◦ c3 ) + α2 β2 γ2 (a1 ◦ b1 ◦ c1 ) + α3 β3 γ3 (a2 ◦ b2 ◦ c2 ) ✹✼ 3 aj ◦ b j ◦ c j = ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ j=1 ❱➻ ✈➟② aj ◦ b j ◦ c j ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ αj βj γj = 1, j = 1, 2, 3✳ j=1 A, B, C = (A, B, C) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ Ta Tb Tc = I3 ✳ ✣è✐ ✈ỵ✐ ♠ỉ ❤➻♥❤ ❚✉❝❦❡r✸✱ ❝❤ó♥❣ t❛ ❜✐➳t r➡♥❣ ❝→❝ ♠❛ tr➟♥ ❚✉❝❦❡r✸ ❦❤æ♥❣ ❞✉② ♥❤➜t✳ ❚❤➟t ✈➟②✱ ♥➳✉ S ∈ Rq×q ✱ V ∈ Rr×r ✈➔ W ∈ Rs×s A✱ B ✱ C ✈➔ G ❧➔ ❝→❝ ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥ t❤➻ AG (C ⊗ B)T = AG C T ⊗ B T = A ST −1 S T G (W ⊗ V ) W −1 ⊗ V −1 = A ST −1 S T G (W ⊗ V ) W −1 C T ⊗ V −1 B T C T ⊗ BT ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❝❤ó♥❣ t❛ ❝â t❤➸ q✉❛② ❝→❝ ♠❛ tr➟♥ t❤➔♥❤ ♣❤➛♥ ✈➔ ♠↔♥❣ ❝♦r❡ tø A✱ B ✱ C ❧➛♥ ❧÷đt s❛♥❣ A ST −1 ✱ B VT −1 ✈➔ C WT −1 ✳ ▼ët ♣❤➨♣ q✉❛② ①✐➯♥ ✤➸ ❝â ✤÷đ❝ ❝➜✉ tró❝ ✤ì♥ ❣✐↔♥ tr♦♥❣ ♠↔♥❣ s❝♦r❡ ❝â t❤➸ ✤÷đ❝ t➻♠ t❤➜② tr♦♥❣ ❑✐❡rs ✭✶✾✾✽❜✮✳ ▼ët ♣❤➨♣ q✉❛② trü❝ ❣✐❛♦ ✤➸ ❝â ✤÷đ❝ ❝➜✉ tró❝ ✤ì♥ ❣✐↔♥ tr♦♥❣ ♠↔♥❣ s❝♦r❡ ❝â t❤➸ ✤÷đ❝ t➻♠ t❤➜② tr♦♥❣ ❑✐❡rs ✭✶✾✾✽❛✮✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ♠ỉ ❤➻♥❤ ❈P ✤➳♥ ♥❛② ❤➛✉ ♥❤÷ ✤➣ ✤÷đ❝ ❣✐↔✐ q✉②➳t ♠ët ❝→❝❤ trå♥ ✈➭♥✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❞✉② ♥❤➜t ❝õ❛ ♠æ ❤➻♥❤ ❚✉❝❦❡r✸ ❝❤➾ ✤÷❛ r❛ ❝❤♦ ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ♥➔♦ ✤â✳ ✹✽ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉✿ ✶✳ ❚➻♠ ❤✐➸✉ ✈➲ ❤❛✐ ♠æ ❤➻♥❤ ♣❤➛♥ t➼❝❤ ❤❛✐ ❝❤✐➲✉✱ ✤â ❧➔ P❤➙♥ t➼❝❤ ❣✐→ trà ❦➻ ❞à ✭❙❱❉✮ ✈➔ P❤➙♥ t➼❝❤ t❤➔♥❤ ♣❤➛♥ ❝❤➼♥❤ ✭P❈❆✮✱ ✈➔ ✤÷❛ r❛ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ❤❛✐ ♠æ ❤➻♥❤ ♥➔②✳ ✷✳ ◆❣❤✐➯♥ ❝ù✉✱ tr➻♥❤ ❜➔② ❧↕✐ ❤❛✐ ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ❜❛ ❝❤✐➲✉✱ ✤â ❧➔ ▼æ ❤➻♥❤ ❈P ✈➔ ▼æ ❤➻♥❤ ❚✉❝❦❡r✸✱ ✈➔ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❤❛✐ ♠ỉ ❤➻♥❤ ♥➔②✳ ▼➦❝ ❞ị ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ s➩ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ ữủ ỳ ỵ õ õ ỵ ổ ữủ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✹✾ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t✿ ❬✶❪ ◆❣✉②➵♥ ❉✉② ❚❤✉➟♥✱ P❤✐ ▼↕♥❤ ❇❛♥✱ ◆æ♥❣ ◗✉è❝ ❈❤✐♥❤✱ ✣↕✐ sè t✉②➳♥ t➼♥❤✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✭✷✵✵✸✮✳ ❬✷❪ ❱ơ ❍ú✉ ❚✐➺♣✱ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❝ì ❜↔♥✱ ◆❤➔ ①✉➜t ❜↔♥ ❑❤♦❛ ❤å❝ ✈➔ ❑ÿ t❤✉➟t✱ ✭✷✵✶✽✮✳ ❚✐➳♥❣ a ăr Pr t ss s ♠❛t❡r✐❛❧ ❢r♦♠ t❤❡ ❜♦♦❦ ◆❛t✉r❛❧ ■♠❛❣❡ ❙t❛t✐st✐❝s t♦ ❜❡ ♣✉❜❧✐s❤❡❞ ❜② ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✐♥ ✷✵✵✾✱ ✷✵✵✾✳ ❬✹❪ ❈❛r❧ ❉✳ ▼❡②❡r✱ ▼❛tr✐① ❛♥❛❧②s✐s ❛♥❞ ❛♣♣❧✐❡❞ ❧✐♥❡❛r❛✱ ❙■❆▼✱ P❤✐❧❛❞❡❧♣❤✐❛ ✷✵✵✵✳ ❬✺❪ ❈❛r♦❧❧ ❏✳ ❉✳ ❛♥❞ ❈❤❛♥❣ ❏✳ ❏✱ ❆♥❛❧②s✐s ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❞✐❢❢❡r❡♥❝❡s ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ s❝❛❧✐♥❣ ✈✐❛ ❛♥ tr✐❦❛✱ n✲✇❛② ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❊❝❦❛rt✲❨♦✉♥❣ ❞❡❝♦♠♣♦s✐t✐♦♥✱ Ps②❝❤♦♠❡✲ ✸✺ ✭✸✵✮✱ ♣✳✷✽✸✲✸✶✾✱ ✶✾✼✵✳ ❬✻❪ ❉❛✈✐❞ ❈✳ ▲❛②✱ ▲✐♥❡❛r ❛❧❣❡❜r❛ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s✱ ❆❞❞✐s♦♥✲❲❡s❧❡②✱ ❘❡❛❞✐♥❣✱ ▼❆✱ ✶✾✾✹✳ ❬✼❪ ❍❛rs❤♠❛♥ ❘✳ ❆✳✱ ❋♦✉♥❞❛t✐♦♥s ♦❢ P❛r❛❢❛❝ ♣r♦❝❡❞✉r❡✿ ▼♦❞❡❧s ❛♥❞ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥ ✧❡①♣❧❛♥❛t♦r②✧ ♠✉❧t✐♠♦❞❛❧ ❢❛❝t♦r ❛♥❛❧②s✐s✱ ❯❈▲❆ ❲♦r❦✐♥❣ ♣❛♣❡rs ✐♥ P❤♦♥❡t✐❝s✱ ✶✻✱ ♣✳✶✲✽✹✱ ✶✾✼✵✳ ❬✽❪ ■✳ ❏✳ ●♦♦❞✱ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♠❛tr✐① ❚❡❝❤✲ ♥♦♠❡tr✐❝s✱ ✶✶✱ ♣✳✽✷✸ ✲ ✽✸✶✱ ✶✾✻✾✳ ✺✵ ❬✾❪ ❑r✉s❦❛❧ ❏✳ ❇✳✱ ❘❛♥❦✱ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ✉♥✐q✉❡♥❡ss ❢♦r t❤r❡❡✲✇❛② ❛♥❞ N ✲✇❛② ❛r✲ r❛②s✱ ■♥ ❘✳ ❈♦♣♣✐ ✫ ❙✳ ❇♦❧❛ss❝♦ ✭❊❞s✳✮✱ ▼✉❧t✐✇❛② ❞❛t❛ ❛♥❛❧②s✐s✱ ❊❧s❡✈✐❡r ❙❝✐❡♥❝❡ P✉❜❧✐s❤❡r ❇✳❱✳ ✭◆♦rt❤ ❍♦❧❧❛♥❞✮✱ ✶✾✽✾✳ ❬✶✵❪ ▲❛♠ ❚❤✐ ❚❤❛♥❤ ❚❛♠✱ ❙♦♠❡ ♥❡✇ ♠❡t❤♦❞s ❢♦r t❤r❡❡✲♠♦❞❡ ❢❛❝t♦r ❛♥❛❧②s✐s ❛♥❞ ♠✉❧t✐✲ s❡t ❢❛❝t♦r ❛♥❛❧②s✐s✱ P❤❉ ❚❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ●r♦♥✐♥❣❡♥✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s ✷✵✶✺✳ ❬✶✶❪ ❙t❡✇❛rt ●✳ ❲✳✱ ❖♥ t❤❡ ❡❛r❧② ❤✐st♦r② ♦❢ t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ❙■❆▼ ❘❡✈✐❡✇✱ ✸✺✱ ♣✳✺✺✶ ✲ ✺✻✻✱ ✶✾✾✸✳ ❬✶✷❪ ❚❤♦♠❛s ❙✳ ❙❣♦r❡s✱ ❆♣♣❧✐❡❞ ❧✐♥❡❛r ❛❧❣❡❜r❛ ❛♥❞ ♠❛tr✐① ❛♥❛❧②s✐s✱ ❙♣r✐♥❣❡r✱ ✷✵✵✵✳ ❚❤❛♠ ❦❤↔♦ tr➯♥ ■♥t❡r♥❡t✿ ❬✶✸❪ ❲❡❜s✐t❡✿ ❤tt♣s✿✴✴t✉❛♥✈❛♥❧❡✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✶✸✴✶✷✴✷✺✴♣❤✉♦♥❣✲ ♣❤❛♣✲♣❤❛♥✲t✐❝❤✲t❤❛♥❤✲♣❤❛♥✲❝❤✐♥❤✲♣r✐♥❝✐♣❛❧✲❝♦♠♣♦♥❡♥t✲❛♥❛❧②s✐s✲ ♣❝❛✴❄❢❜❝❧✐❞❂■✇❆❘✶◆◆●❙❜❆✺❲❙❛✵❤②❞❱✼❋♠◆✈✾✵③❨✐❣❖❴❖③❇❝❩✵✷③✽✐❳②❖❘❧✾✾ t③❨❤✈❝▲③❖✺s✳ ✺✶ ◗✉②➳t ✤à♥❤ ❣✐❛♦ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ... E, gt (at ◦ bt ◦ ct ) + E = X= t=1 t=1 ✸✾ ✭✸✳✶✮ tr♦♥❣ ✤â E ❧➔ ♠↔♥❣ ❞÷✱ ✤ë ❞➔✐ ❜➡♥❣ gt q số trữợ trồ số ❝õ❛ t❤➔♥❤ ♣❤➛♥ t❤ù ❤↕♥❣ ✶✱ ❣å✐ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥✳ ❈è ✤à♥❤ tè✐ t❤✐➸✉ E at ∈ Rm ✱ bt ∈ Rn... ❍↕♥❣ ❝õ❛ ♠ët ♠❛ tr➟♥ ❜➡♥❣ sè ❝→❝ ❣✐→ trà ❦➻ ❞à ❦❤→❝ ❦❤æ♥❣ ❝õ❛ ♥â✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû X Rmìn ợ số tr ❦❤→❝ ❦❤æ♥❣ ❝õ❛ ✣➦t Ur = u1 u2 ur ✱ m ≥ n✱ ❝â ♣❤➙♥ t➼❝❤ ❙❱❉ ❧➔ X = U SV T ✱ ✈➔ r ❧➔... rank(S) = r ỵ X ❧➔ ♠❛ tr➟♥ ❝ï m × n✳ ●✐↔ sû X õ t ữợ tr X = σ1 u1 v1T + + r ur vrT ợ k số ữỡ t❤ä❛ k ≤ r✱ ✤➦t Xk = σ1 u1 v1T + + σk uk vkT ✳ ❑❤✐ ✤â rank(Xk ) = k ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣÷❛