Tìm hiểu một số dạng chính tắc của các ma trận đặc biệt như ma trận Hermit, ma trận đường chéo, ma trận tự liên hợp trong không gian phức, mở rộng các kết quả đã biết trong không gian thực đã được biết đến trong bộ môn Đại số tuyến tính và Hình học giải tích.
▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ữợ tr Cn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❈❤÷ì♥❣ ✷✳ ▼ët sè ❞↕♥❣ ♠❛ tr➟♥ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳ ▼❛ tr➟♥ H ✲ tü ❧✐➯♥ ❤ñ♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ▼❛ tr➟♥ H− ✉♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳ ❚÷ì♥❣ ✤÷ì♥❣ ❯♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳ ▼❛ tr➟♥ ❙②♠♣❧❡❝t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✾ ✷✹ ✷✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶ ▲í✐ ♥â✐ ✤➛✉ ❚r♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠❛ tr➟♥✱ ❝ơ♥❣ ♥❤÷ ✈✐➺❝ tr♦♥❣ t➼♥❤ t♦→♥ ❝→❝ ❤➔♠ ♠❛ tr➟♥✱ ✈✐➺❝ ❧➔♠ t♦→♥ trü❝ t✐➳♣ r➜t ❦❤â ❦❤➠♥✳ ❱➻ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ✤÷❛ ❝→❝ ♠❛ tr➟♥ ✈➲ ❞↕♥❣ ❝❤➼♥❤ t➢❝ trð ♥➯♥ r➜t q✉❛♥ trồ tữỡ ữỡ ợ ự t r❛ ❞↕♥❣ ✤ì♥ ❣✐↔♥ ♥❤➜t ❝õ❛ ♠❛ tr➟♥ ✭❞↕♥❣ ❏♦r❞❛♥✮ ♥❣❤➽❛ ❧➔ t➻♠ r❛ ♠ët ♠❛ tr➟♥ ✤ì♥ ❣✐↔♥ ♥❤➜t ỗ ợ õ tỹ q ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✈➲ ❣✐→ trà r✐➯♥❣✱ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣✳ ✣è✐ ✈ỵ✐ ♥❤â♠ ❝→❝ ♠❛ tr➟♥ t❤ü❝✱ t❛ ✤➣ t➻♠ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ✤➦❝ ❜✐➺t ❝❤♦ ♠❛ tr➟♥ trü❝ ❣✐❛♦✱ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳✳✳ ❱➟② t❤➻ ♥❤ú♥❣ ❦➳t q✉↔ ♥➔② ❝â t❤➸ ♠ð rë♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ♠❛ tr➟♥ ♣❤ù❝ ❤❛② ❦❤æ♥❣✳ ❑❤â❛ ❧✉➟♥ ✧ ✳✳✳✧ ợ ữỡ t ❝➜✉ tró❝ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣✱ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠ët sè ❧ỵ♣ ❝→❝ ♠❛ tr➟♥ ♣❤ù❝ ✤➦❝ ❜✐➺t✳ ❈ư t❤➸ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ✈➲ ❣✐→ trà r✐➯♥❣✱ tự trữ ởt ỗ tr ❧ơ② ❧✐♥❤ ✈➔ ♠❛ tr➟♥ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥✳ ❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤ê ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ët sè ♠❛ tr➟♥ ♠❛ tr➟♥ ✤➦❝ ❜✐➺t ♥❤÷✿ ♠❛ tr➟♥ H ✲ tü ❧✐➯♥ ❤đ♣✱ ♠❛ tr➟♥ ❯♥✐t❛✳✳✳ ❚r♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❤❙✳ q t ữợ ữ r ỵ õ õ tổ ❝â t❤➸ ❤♦➔♥ t❤✐➺♥ ♥ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ò♥❣ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥✱ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✷ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❙✐♥❤ ✈✐➯♥ ❚r➛♥ ❚❤à ỗ ữỡ ởt số tự ❜à ❚❛ ❧✉æ♥ ❣✐↔ sû V ❧➔ K− ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ f ∈ End(V )✳ ✶✳✶✳ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ U ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ V ✳ U ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ f ✭❤❛② f − ❜➜t ❜✐➳♥✮ ♥➳✉ f (U ) ⊂ U ✳ ✣✐➲✉ ❦✐➺♥ f (U ) ⊂ U t÷ì♥❣ ✤÷ì♥❣ ∀ε ∈ U → f (ε) ∈ U ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â✿ ✶✳ ◆➳✉ U ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ f t❤➻ f s ởt tỹ ỗ f|U : U → U ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ f ❧➯♥ U − − ✷✳ ❈❤å♥ ♠ët ❝ì sð {→ ε1 , , , m } U rỗ s✉♥❣ ✤➸ ✤÷đ❝ ❝ì sð {ε1 , , εm , εm+1 , , εn } ❝õ❛ V ✳ ❱➻ f (εi ), i = 1, , m ❝❤➾ ❜✐➸✉ t❤à q✉❛ ❤➺ {ε1 , ε2 , , εm } ♥➯♥ tr♦♥❣ ❝ì sð {ε1 , , εm , εm+1 , , εn } ❝õ❛ V ✱ f ❝â ♠❛ tr➟♥ ❝â ❞↕♥❣ A B O C ð ✤➙② A ❧➔ ♠❛ tr➟♥ ❝õ❛ f|U ✤è✐ ✈ỵ✐ ❝ì sð {ε1, ε2, , εm}✱ O ❧➔ ♠❛ tr➟♥ ❦❤ỉ♥❣✳ ❍ì♥ ♥ú❛✱ ✈➻ ✤✐➲✉ ❦✐➺♥ f (U ) ⊂ U ♥➯♥ f ❝á♥ s ởt tỹ ỗ fU : V /U → V /U ✱ fU ([α]) = [f (α)]✳ ❑❤✐ ✤â✱ C ❝❤➼♥❤ ❧➔ ♠❛ tr➟♥ ❝õ❛ f¯U ✤è✐ ợ ỡ s [m+1], , [n] tỗ t ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❜➜t ❜✐➳♥ W ❦❤→❝ ❝õ❛ V t❤ä❛ ♠➣♥ V = U ⊕ W ✈➔ t❛ ❝❤å♥ {εm+1 , , εn } ❧➔ ❝ì sð ❝õ❛ W t❤➻ ♠❛ tr➟♥ ❝õ❛ ✹ f tr♦♥❣ ❝ì sð {ε1, , εm, εm+1, , εn} ❝â ❞↕♥❣ A O O C ð ✤➙② A ❧➔ ♠❛ tr➟♥ ❝õ❛ f|U ✤è✐ ✈ỵ✐ ❝ì sð {ε1, ε2, , εm}✱ C ❧➔ ♠❛ tr➟♥ ❝õ❛ f|W ✤è✐ ✈ỵ✐ {εm+1, , εn}✱ O ❧➔ ♠❛ tr➟♥ ❦❤ỉ♥❣✳ ❉♦ ✈➟② ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❝õ❛ f t❤➻ t❛ ✤÷❛ ✈➲ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❝õ❛ f|U ✈➔ f|W ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ●✐↔ sû f : V V ởt tỹ ỗ tr➯♥ K ✳ ❱❡❝tì → − − − α = ♠➔ f (→ α ) = λ→ α ✈ỵ✐ ♠ët λ ♥➔♦ ✤â ❝õ❛ tr÷í♥❣ K ✱ ❣å✐ ❧➔ ✈❡❝tì r✐➯♥❣ ❝õ❛ f ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ ❝õ❛ f ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ✣❛ t❤ù❝ ✤➦❝ tr÷♥❣ tỹ ỗ f ỵ Pf (x)✱ ❧➔ ✤❛ t❤ù❝ Pf (X) = det(f − xIdV ) ✳ Pf (x) ❧➔ ♠ët ✤❛ t❤ù❝ ❜➟❝ n ✭❝â ❤➺ sè ❝❛♦ ♥❤➜t ❧➔ (−1)n ✮✳ ●✐↔ sû tr♦♥❣ ❝ì sð ε = {εi} ❝õ❛ V ✱ f ❝â ♠❛ tr➟♥ ❧➔ A t❤➻ Pf (λ) = det(f − λIdV ) = det(A − λIn ) = PA (lambda|)✳ ✣❛ t❤ù❝ PA (λ) ❝ô♥❣ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♠❛ tr➟♥ A ❳➨t ✤❛ t❤ù❝ p(x) = amxm + + a1x + a0 tr➯♥ tr÷í♥❣ K✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ✲ ▼❛ tr➟♥ ✈✉æ♥❣ A ❝➜♣ n tr➯♥ K ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ p(x) ♥➳✉ p(A) = am Am + + a1 A + a0 In = O ✭O ❧➔ ♠❛ tr➟♥ ổ ởt tỹ ỗ f End(V ) ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ P (X) ♥➳✉ p(f ) = am f m + + a1 f + a0 IdV = 0 tỹ ỗ ❝➜✉ ❦❤ỉ♥❣✳✮ ❱ỵ✐ λ ∈ K ✱ ①➨t Ker(f − λIdV )✳ ❑❤✐ ♥â ❦❤→❝ t❤➻ ✤â ❧➔ ❦❤æ♥❣ tỡ V ỗ tỡ tt ❝↔ ❝→❝ ✈❡❝tì r✐➯♥❣ ❝õ❛ f ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ✳ ❑❤æ♥❣ ❣✐❛♥ ♥➔② ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ ự ợ tr r ữủ ❧➔ Pλ✳ ❱➟② Pλ = Ker(f − λIdV ) ✺ sỷ f ởt tỹ ỗ ❝➜✉ ❝õ❛ K− ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì − V ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❱ỵ✐ ♠é✐ λ ∈ K ✱ ①➨t t➟♣ {→ α ∈ V | ❝â sè ♥❣✉②➯♥ ❞÷ì♥❣ ♠ ✤➸ − (f − λIdV )m (→ α ) = 0} ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ V ✳ ❑❤✐ ♥â ❦❤→❝ t❤➻ ❣å✐ ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ ❝õ❛ f ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ ✈➔ ❦➼ ❤✐➺✉ ❧➔ Rλ✳ ▼➺♥❤ ✤➲ ✶✳✶✳✶✳ ❈❤♦ Rλ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ ❝õ❛ f ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ✱ t❛ ❝â ♥❤ú♥❣ ♠➺♥❤ ✤➲ s❛✉✿ ✶✳ Rλ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ f ỹ ỗ f |R : R → Rλ ❝❤➾ ❝â ❣✐→ trà r✐➯♥❣ ❧➔ λ✳ ✸✳ ❙è ❝❤✐➲✉ ✤↕✐ sè ❝õ❛ λ ❜➡♥❣ ❜ë✐ ❝õ❛ ♥❣❤✐➺♠ λ ❝õ❛ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ Pf (x)✳ → − − ✹✳ ◆➳✉ → α k ∈ Rλk \ { }✱ k = 1, 2, , m ♠➔ λ1 , λ2 , , λm ✤æ✐ ♠ët ♣❤➙♥ − − − ❜✐➺t t❤➻ ❤➺ ✈❡❝tì {→ α 1, → α , , → α m } ✤ë❝ ❧➟♣ t✉②➳♥ t ỹ ỗ f ❧➔ ❧ơ② ❧✐♥❤ ♥➳✉ ❝â sè ♥❣✉②➯♥ ❞÷ì♥❣ q s❛♦ ❝❤♦ f q = 0✱ ✭f q = f◦f ◦ ◦ f ✱ q ❧➛♥ ✮✳ ❚❤➯♠ ✈➔♦ ✤â ♥➳✉ q ❧➔ sè ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ f q = t❤➻ q ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❧ơ② ❧✐♥❤ ❝õ❛ f ✳ − − − − ✷✳ ▼ët ❝ì sð {→ ε , , → ε n } ❝õ❛ V ♠➔ f (→ ε i) = → ε i+1 (i = 1, , n − 1) ✈➔ → − → − f ( ε n ) = ữủ ỡ s ố ợ f − − ✸✳ ▼ët ❝ì sð {→ ε , , → ε m } ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ U V ữủ ố ợ f ♥➳✉ f (εi) = εi+1(i = 1, , m − 1) ✈➔ f (εm) = → 0✳ ✹✳ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ U ❝õ❛ V ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f ♥➳✉ ♥â ❝â ♠ët ❝ì sð ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ t❤➜② r➡♥❣ ♥➳✉ U ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f t❤➻ U ❧➔ f − ❜➜t ❜✐➳♥✳ ❍ì♥ ♥ú❛ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ✶✳ ▼å✐ ❣✐→ tr r ởt tỹ ỗ ụ ❧➔ ❦❤æ♥❣✳ ✻ − − ✷✳ ◆➳✉ {→ ε , , → ε n } ❧➔ ❝ì sð ①②❝❧✐❝ ❝õ❛ f tr♦♥❣ ❝ì sð ♥➔② ❝â ❞↕♥❣ 1 0 0 0 :V →V t❤➻ ♠❛ tr➟♥ ❝õ❛ f 0 0 ỵ s t trú ởt tỹ ỗ ❧ơ② ❧✐♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ✣à♥❤ ỵ f tỹ ỗ ụ K✲ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ V t❤➻ V ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ♥❤ú♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ố ợ f rữợ ự ỵ t ởt số t q s t Vi = Kerf i ✱ i ≥ ✭f = IdV ✮✳ ❚❛ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ Vi ⊆ Vi+1 ✈➔ f (Vi+1 ) ⊆ Vi ✳ ❍ì♥ ♥ú❛ ▼➺♥❤ ✤➲ ✶✳✶✳✷✳ ◆➳✉ q ❧➔ ❜➟❝ ❧ô② ❧✐♥❤ ❝õ❛ f t❤➻ Vi Vi+1 ✈ỵ✐ ♠å✐ i < q ✳ ự ự tỗ t i s ❝❤♦ Vi = Vi+1 t❤➻ Vi = → − − − Vi+1 = = Vq = V ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ → α ∈ Vi+2 t❤➻ f i+2 (→ α) = ✱ → − → − − − − s✉② r❛ f i+1(f (→ α )) = =⇒ f (→ α ) ∈ Vi+1 = Vi ✱ s✉② r❛ f i (→ α ) = tù❝ → − α ∈ Vi+1 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭❝õ❛ ỵ (q) (q) Vq = V s❛♦ ❝❤♦ ❤➺ {[→ α ], , [→ α rq ]} ❧➔ (q) − f (→ α j )✳ ❚❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ (q) (q) − − ❝→❝ ✈❡❝tì {→ α , , → α r } tr♦♥❣ (q−1) − ❝ì sð ❝õ❛ V /Vq−1✳ ✣➦t → αj = q (q) (q) − − ✶✳ ❍➺ {→ α , , → α r } ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr♦♥❣ V ✳ (q−1) (q−1) − − ✷✳ ❍➺ {[→ α ], , [→ α r ]} ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr♦♥❣ Vq−1 /Vq−2 ✳ ❑❤➥♥❣ ✤à♥❤ ✶ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤ ✷✳ ❳➨t ✤➥♥❣ t❤ù❝✿ q q → − (q−1) − − k1 [→ α ] + + krq [→ α (q−1) ]=[0] rq ✼ (q−1) (q−1) − − tù❝ ❧➔ k1f (→ α ) + + krq f (→ α rq ) ∈ Vq−2 ⊂ Vq−1 ✳ ❙✉② r❛ rq j=1 (q) − kj [→ αj ] = tr♦♥❣ Vq /Vq−1✳ ❙✉② r❛ kj = ✈ỵ✐ ♠å✐ j = 1, , rq ✳ (q−1) (q−1) − − ❚ø ❤❛✐ ❦❤➥♥❣ ✤à♥❤ tr➯♥ t❛ ❝â rq ≤ rq−1✳ ❇ê s✉♥❣ → α r +1 , , → αr t❤➔♥❤ (q−1) (q−1) − (q−1) (q−1) (q−1) (q−1) − (q−1) − − − − − ❤➺ {→ α , , → αr ,→ α r +1 , , → α r } s❛♦ ❝❤♦ ❤➺ {[→ α ], , [→ α r ], [→ α r +1 ], ❧➔ ❝ì sð ❝õ❛ Vq−1/Vq−2✳ ❚✐➳♣ tư❝ q✉→ tr➻♥❤ ♥❤÷ tr➯♥ t❛ t❤✉ ✤÷đ❝ ❝→❝ ✈❡❝tì → − [0] q q q−1 q−1 q q (q) → − − α , , → α (q) rq (q−1) (q−1) → − − − − α , , → α (q−1) , → α , , → α (q−1) rq rq +1 rq−1 (1) − → − (1) → − → − (1) → − (1) → − (1) α (1) α , , → rq , α rq +1 , , α rq−1 , , α r2 , , α r1 s❛♦ ❝❤♦ ợ tữỡ ữỡ tỡ ỏ tự i tứ ữợ ỡ s ❝õ❛ Vi/Vi−1✳ ❍ì♥ ♥ú❛✱ f (→ α ji ) = j1 i t ỗ tt ❝↔ ❝→❝ ✈❡❝tì tr➯♥ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ❧➔ ♠ët ❝ì sð ❝õ❛ V ✳ ✣➦t Ui ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ V s✐♥❤ ❜ð✐ ❝→❝ ✈❡❝tì ð ❝ët t❤ù i✳ ❉➵ t❤➜② Ui ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f ✳ ❱➔ V = U1 ⊕ Ur ứ ự ỵ t t❤➜② ✶✳ ◆➳✉ f ∈ End(V ) ✈➔ f ❧ô② ❧✐♥❤ t❤➻ ❝â ♥❤✐➲✉ ❝→❝❤ ♣❤➙♥ t➼❝❤ V t❤➔♥❤ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f ✳ ❚✉② ♥❤✐➯♥ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ s ≥ t❤➻ sè ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ①②❝❧✐❝ s ❝❤✐➲✉ ✤è✐ ✈ỵ✐ f ✤➲✉ ❜➡♥❣ ♥❤❛✉ ✈➔ ❝ư t❤➸ ❧➔ ❜➡♥❣ ❤↕♥❣f s−1 − 2❤↕♥❣f s + ❤↕♥❣f s+1 f tỹ ỗ ụ f t❤➻ V ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ①②❝❧✐❝ ✤è✐ ✈ỵ✐ f ✈➔ tr♦♥❣ ♠é✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ①②❝❧✐❝ ✤â ❝❤å♥ ❝ì sð ①②❝❧✐❝ t❤➻ ♠❛ tr➟♥ ❝õ❛ f s➩ ❝â ✽ q ❞↕♥❣ s❛✉ ✭❣å✐ ❧➔ ♠❛ tr➟♥ ❞↕♥❣ ❝❤✉➞♥ t➢❝ tỹ ỗ ụ A1 A2 O O , ✭✶✳✶✳✶✮ Ak tr♦♥❣ ✤â ♠é✐ Ai ♥➡♠ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ✤➲✉ ❝â ❞↕♥❣✿ 1 0 0 0 0 . 0 ●✐↔ sû λ ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â (f IdV )|R ởt tỹ ỗ ụ õ tỗ t ởt ỡ s Rλ s❛♦ ❝❤♦ ♠❛ tr➟♥ ❝õ❛ (f −λIdV )|R ✤è✐ ợ ỡ s õ õ ữ f = (f −λIdV )+λIdV ♥➯♥ t❛ ❝â✱ tr♦♥❣ ❝ì sð tr➯♥ t❤➻ f |R ❝â ♠❛ tr➟♥ ❞↕♥❣ λ λ λ Aλ,1 Aλ,2 O O , ✭✶✳✶✳✷✮ Aλ,k tr♦♥❣ ✤â ♠é✐ Aλ,i ♥➡♠ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ✤➲✉ ❝â ❞↕♥❣ λ 1 0 0 λ 0 λ 0 . 0 λ ❙è ❦❤✉♥❣ ❝➜♣ s ❜➡♥❣ ❤↕♥❣(f − λIdV )s−1 − 2❤↕♥❣(f − λIdV )s + ❤↕♥❣(f − λIdV )s+1 ỵ q trồ s t t trú ởt tỹ ỗ tr ổ ỳ ỵ s r trỹ t tứ t t tr ỵ f ởt tỹ ỗ ổ tỡ ỳ ❤↕♥ ❝❤✐➲✉ V tr➯♥ tr÷í♥❣ K ♠➔ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ Pf (X) ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ ❝→❝ ♥❤➙♥ tû t✉②➳♥ t➼♥❤ Pf (x) = (−1)n (x − λ1 )s1 (x − λ2 )s2 (x − λm )sm ✭❝→❝ λi ❦❤→❝ ♥❤❛✉ tø♥❣ ✤æ✐ ♠ët ✮ t❤➻ V ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ ❝õ❛ f V = Rλ1 ⊕ Rλ2 ⊕ ⊕ Rλm ✈➔ ❞♦ ✤â V ❝â ❝ì sð ✤➸ ♠❛ tr➟♥ ❝õ❛ f tr♦♥❣ ❝ì sð ✤â t↕♦ ❜ð✐ ♥❤ú♥❣ ❦❤✉♥❣ ❏♦r❞❛♥ Aλ1 Aλ2 O ✭✶✳✶✳✸✮ , O Aλk tr♦♥❣ ✤â ♠é✐ Aλi ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ si ❝â ❞↕♥❣ ♥❤÷ ✭✶✳✶✳✷✮ ♥➡♠ ❞å❝ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤✳ ❙è ❦❤✉♥❣ ❏♦r❞❛♥ ❝➜♣ s ✈ỵ✐ ♣❤➛♥ tû ❝❤➨♦ ✭❣✐→ trà r✐➯♥❣✮ λk ❜➡♥❣ ❤↕♥❣(f − λk IdV )s−1 − 2❤↕♥❣(f − λk IdV )s + ❤↕♥❣(f − λk IdV )s+1 ✱ ♥➯♥ ♠❛ tr➟♥ ✤â ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ f s❛✐ ❦❤→❝ ❝→❝❤ s➢♣ ①➳♣ ❝→❝ ❦❤✉♥❣ ❏♦r❞❛♥ ❞å❝ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ✈➔ ❣å✐ ❧➔ ♠❛ tr➟♥ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥ ❝õ❛ f✳ ❈❤✉②➸♥ s❛♥❣ ❦➳t q✉↔ ❝❤♦ ♠❛ tr➟♥ t❛ ✤÷đ❝ ❍➺ q✉↔ ✶✳✶✳✶✳ A ❧➔ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n tr÷í♥❣ K ♠➔ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ PA (x) ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ ❝→❝ ♥❤➙♥ tû t✉②➳♥ t➼♥❤ Pf (x) = (−1)n (x − λ1 )s1 (x − λ2 )s2 (x − λm )sm ✭❝→❝ λi ❦❤→❝ ♥❤❛✉ tứ ổ ởt t tỗ t ởt tr ✈✉æ♥❣ ❝➜♣ n ❦❤↔ ♥❣❤à❝❤ C s❛♦ ❝❤♦ C −1 AC ❝â ❞↕♥❣ ♥❤÷ ✭✶✳✶✳✸✮ ❚❛ ❝â t❤➸ ♥❤➟♥ t❤➜② ♠å✐ ♠❛ tr➟♥ tr➯♥ C ✤➲✉ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ♥❤÷ ✈➟②✳ ✶✵ ▼➺♥❤ ✤➲ ✷✳✶✳✷✳ ❈❤♦ A : Cn → Cn t ổ ữợ tr Cn ✤â✱ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❧➔ A− ❜➜t ❜✐➳♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ M [⊥] ❧➔ A[∗] − ❜➜t ❜✐➳♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② M ❧➔ A− ❜➜t ❜✐➳♥ ✈➔ x ∈ M ✱ y ∈ M[⊥]✳ ❑❤✐ ✤â✿ [A[∗] y, x] = [y, Ax] = 0, tø ✤â Ax t❤✉ë❝ M✳ ❱➟② A[∗]y ∈ M[⊥] ✈➔ M[⊥] ❧➔ A[∗]− ❜➜t ❜✐➳♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐✱ t❛ →♣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ ❝❤♦ A[∗] ✈➔ (A[∗] )[∗] = A ❝ò♥❣ ✈ỵ✐ t➼♥❤ ❝❤➜t (M[⊥] )[⊥] = M✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✸✳ ❈❤♦ A ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n✳ ❑❤✐ ✤â✱ A ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤ñ♣ ✭❤❛② tü ❧✐➯♥ ❤ñ♣ ✈ỵ✐ ❬✳✱✳❪✮ ♥➳✉ A = A[∗]✳ ▼➺♥❤ ✤➲ ✷✳✶✳✸✳ ❈❤♦ H1 H2 t ổ ữợ tr C n ✈➔ H2 = SH1 S ∗ ✈ỵ✐ S ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ❝➜♣ n✳ ❑❤✐ ✤â A1 ❧➔ H1 ✲ tü ❧✐➯♥ ❤ñ♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠❛ tr➟♥ A2 := (S ∗ )−1 A1 S ∗ ❧➔ H2 ✲ tü ❧✐➯♥ ❤ñ♣✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝õ❛ ♠➺♥❤ ✤➲✳ ❈❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✤↔♦ ❧➔ t÷ì♥❣ tü✳ ●✐↔ sû r➡♥❣ A1 ❧➔ H1− tü ❧✐➯♥ ❤ñ♣✱ tø ✭✷✳✶✳✶✮✱ t❛ ❝â✿ H1A1 = A∗1H1✳ ❑❤✐ ✤â ❈❤ù♥❣ ♠✐♥❤✳ H2 A2 = (SH1 S ∗ )((S ∗ )−1 A1 S ∗ ) = SH1 A1 S ∗ = SA∗1 H1 S ∗ = (SA∗1 S −1 )(SH1 S ∗ ) = A∗2 H2 s✉② r❛ A2 ❧➔ H2− tü ❧✐➯♥ ❤ñ♣✳ ❈❤♦ ❬✳✱✳❪❂✭❍✳✱✳✮ ❧➔ ♠ët t➼❝❤ ổ ữợ tr C n t ởt tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ H− tü ❧✐➯♥ ❤đ♣ ♥➳✉ A = A[∗] ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝✿ A = H −1 A∗ H ✭✷✳✶✳✺✮ ❉♦ ✤â✱ ❜➜t ❦➻ ♠❛ tr➟♥ H tỹ ủ A ụ ỗ ợ A ú ỵ r t tt ❞↕♥❣ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤đ♣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝✱ ♥❣❤➽❛ ❧➔ ♥➳✉ A ✈➔ B ❧➔ H− tü ❧✐➯♥ ❤ñ♣ t❤➻ αA + βB ❧➔ H− tü ❧✐➯♥ ❤đ♣ ✈ỵ✐ α, β ❧➔ ❝→❝ sè t❤ü❝ ❜➜t ❦➻✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ H = I t❤➻ ❞➵ ❞➔♥❣ ♥❤➟♥ t❤➜② A ❧➔ H− tü ❧✐➯♥ ❤ñ♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A∗ ❧➔ H− tü ❧✐➯♥ ❤ñ♣✳ ứ õ t õ ú ỵ H = I ✈➔ H ❧➔ ❍❡❝♠✐t t❤➻ A ❧➔ H− tü ủ ỗ tớ i(TA) = (A − A∗ ) RA = (A + A∗ ), ✭✷✳✶✳✻✮ ❧➔ H− tü ❧✐➯♥ ❤ñ♣✳ ❙❛✉ ✤➙② ❧➔ ✈➼ ❞ư ❝ì ❜↔♥ ✈➲ H− tü ❧✐➯♥ ❤đ♣✳ ❱➼ ❞ư ✷✳✶✳✶✳ ❈❤♦ [x, y] = ( Snx, y)✱ x, y ∈ C n✱ tr♦♥❣ ✤â Sn ❧➔ ♠❛ tr➟♥ ✤÷đ❝ ❝❤♦ tr♦♥❣ ♠ư❝ ✭✶✳✷✳✹✮ ✈➔ ♥❤➟♥ ❣✐→ trà ❧➔ ✶ ❤♦➦❝ ✲✶✳ ◆❣♦➔✐ r❛✱ ❝❤♦✿ α 0 J = ✳✳ 0 α ✳✳ ✳✳ 0 α 0 0 ✳✳ 1 α ❧➔ ❦❤✉♥❣ ❏♦r❞❛♥ ❝➜♣ n ✈ỵ✐ ❣✐→ trà r✐➯♥❣ t❤ü❝ α✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝✿ ( Sn )J = J T ( Sn )✱ tø ✤â J T = J ∗ ✤✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ J ❧➔ Sn − tü ❧✐➯♥ ❤ñ♣✳ ❱➼ ❞ö ✷✳✶✳✷✳ ❈❤♦ [x, y] = (Qx, y)✱ x, y ∈ C 2n tr♦♥❣ ✤â✿ Q= ▲➜② K= Sn Sn J , J tr♦♥❣ ✤â J ❧➔ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ ❦❤ỉ♥❣ t❤ü❝ α ✭✈➻ ✈➟② J ❧➔ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ α✮✳ ❉➵ t❤➜②✱ QK = K ∗Q✱ ♥➯♥ K ❧➔ Q− tü ❧✐➯♥ ❤đ♣✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤ñ♣✳ ✶✼ ▼➺♥❤ ✤➲ ✷✳✶✳✹✳ ❈❤♦ A ❧➔ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤ñ♣✳ õ t (A) ỗ tt trà r✐➯♥❣ ❝õ❛ A ✤è✐ ①ù♥❣ q✉❛ trö❝ t❤ü❝✱ tù❝ ❧➔ λ0 ∈ σ(A) ❦➨♦ t❤❡♦ λ0 ∈ σ(A)✳ ❍ì♥ ♥ú❛✱ ✤è✐ ✈ỵ✐ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥✱ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ0 ❜➡♥❣ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ0 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ✭✷✳✶✳✺✮✱ t❛ ❝â✿ λI − A = H −1 (λI − A)∗ H ◆➯♥ λI − A ❧➔ s✉② ❜✐➳♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ λI − A ❧➔ s✉② ❜✐➳♥✱ tù❝ ❧➔✿ λ0 ∈ σ(A) ❦➨♦ t❤❡♦ λ0 ∈ σ(A)✳ ◆❣♦➔✐ r❛✱ ❧➜② J ❧➔ ❞↕♥❣ ❏♦r❞❛♥ ❝õ❛ A ✈ỵ✐ ♠❛ tr➟♥ T s❛♦ ❝❤♦✿ A = T −1JT ú ỵ r J ỗ ợ J ✈➔ t❛ ❧✉ỉ♥ ❝â✿ J ∗ = K −1JK ✈ỵ✐ K ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✳ ❚ø ✤â s✉② r❛✿ λI − A = T −1 (λI − J)T = H −1 {T −1 (λI − J)T }∗ H ❱➟② ♥➯♥✿ λI − J = (T H −1 T ∗ )(λI − J ∗ )((T ∗ )−1 HT −1 ) = S(λI − J)S −1 , tr♦♥❣ ✤â S = T H −1T ∗K −1✳ ❉♦ ✤â✱ J ✈➔ J ỗ J õ t t ữủ tứ J ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ♠ët ✈➔✐ ❦❤✉♥❣ ❏♦r❞❛♥ ❝õ❛ ♥â✳ ữ ỵ r tr r ổ tỹ H tü ❧✐➯♥ ❤ñ♣ ❝❤➾ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝➦♣ ♠❛ tr➟♥ ủ ỵ A ởt tr➟♥ H− tü ❧✐➯♥ ❤đ♣ ✈➔ λ, µ ∈ σ(A) ợ = õ R (A) (Rà (A))[⊥] , tù❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ Rλ (A) Rà (A) trỹ ố ợ [., ] = (H., )✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x ∈ Rλ (A) ✈➔ y ∈ Rµ (A) s❛♦ ❝❤♦ (A − λI)s x = (A − µI)t y = ợ số ữỡ s, t ú t ự r➡♥❣✿ [x, y] = ✶✽ ✈➔ ✭✷✳✶✳✼✮ ❚✐➳♣ tö❝ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ s + t✳ ✣è✐ ✈ỵ✐ s = t = t❛ ❝â ❑❤✐ ✤â✱ = λx✱ Ay = ày ợ = t t❤✉ ✤÷đ❝ ✭✷✳✶✳✼✮✳ ●✐↔ sû r➡♥❣ ✭✷✳✶✳✼✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ợ x R(A) y Rà(A) s (A − λI)s x = (A − λI)t y = ✈ỵ✐ s, t t❤ä❛ ♠➣♥✿ s + t < s + t✳ ▲➜② x, y ♥❤÷ tr➯♥✱ ✤➦t x = (A − λI)x✱ y = (A − µI)y ✳ ❑❤✐ ✤â✱t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✿ [x , y] = [x, y ] = 0✱ s✉② r❛✿ λ[x, y] = [Ax, y]❀ µ[x, y] = [x.Ay]✳ ⑩♣ ❞ư♥❣ ✭✷✳✶✳✽✮ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ λ[x, y] =[Ax, y] = [x, Ay] = [x, µy] = µ[x, y] ✷✳✷✳ ▼❛ tr➟♥ H− ✉♥✐t❛ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n✳ ❑❤✐ ✤â✱ A ✤÷đ❝ ❣å✐ ❧➔ H− ✉♥✐t❛ ♥➳✉ A ❦❤↔ ♥❣❤à❝❤ ✈➔ A−1 = A[∗]✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ A ❧➔ H− ✉♥✐t❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ [Ax, y] = [x, A−1 y] ✈ỵ✐ ♠å✐ x, y ∈ Cn ❤♦➦❝✿ A = H −1 (A∗ )−1 H; A∗ HA =H ✭✷✳✷✳✾✮ ✣➦❝ ❜✐➺t✱ A ỗ ợ (A)1 ữủ ụ ú A ỗ ợ (A)1 t t õ t ởt tr t ú ỵ r ợ ộ H ❝è ✤à♥❤✱ t➟♣ t➜t ❝↔ ❝→❝ H− ✉♥✐t❛ ❧➟♣ t❤➔♥❤ ♥❤â♠✱ tù❝ ❧➔ ♥➳✉ A, B ❧➔ H− ✉♥✐t❛ t❤➻ A−1, B −1 ✈➔ AB ❝ô♥❣ ❧➔ H− ✉♥✐t❛✳ ❱➼ ❞ö ✷✳✷✳✶✳ ▲➜② [x, y] = ( Snx, y)✱ x, y ∈ C n✱ tr♦♥❣ ✤â ❜➡♥❣ ✶ ❤♦➦❝ ✲✶✳ ●✐↔ sû r➡♥❣ λ ∈ C ✈➔ |λ| = ✈➔ λ 2iλ 2i2 λ 2in−2 λ 2in−1 λ λ 2iλ 2in−3 λ 2in−2 λ ✳ ✳ ✳ ✳ ✳ A = ✳ ✳ ✳ ✳ ✳ 0 0 λ 2iλ 0 λ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② r➡♥❣✿ A∗( Sn)A = Sn ♥➯♥ A ❧➔ Sn✲ ✉♥✐t❛✳ ❱➼ ❞ö ✷✳✷✳✷✳ ▲➜② [x, y] = (Qx, y) ✈ỵ✐ ♠å✐ x, y ∈ C n tr♦♥❣ ✤â Q= Sn Sn ✶✾ ❱ỵ✐ λ ∈ C ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ |λ| = ✤➦t✿ A= tr♦♥❣ ✤â✿ λ 0 ✳ K1 = ✳ ✳ ✳ λ−1 ✳ K2 = ✳ ✳ ✳ K1 , K2 k1 k2 kn−1 ✳ λ k1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ , λ k1 λ k1 k2 kn−1 ✳✳ λ−1 k1 ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ λ−1 k1 λ−1 ✈➔ kr = λq1r−1 (q1 − q2 )✱ kr = λ−1 q2r−1 (q2 − q1 ) ✈ỵ✐ r = 1, 2, , n − ✈➔ q1 = 2i (1 + λ)✱ q2 = 2i (1 + λ−1 )✳ ❚❛ t❤➜② ✤÷đ❝ r➡♥❣✿ A∗QA = Q✱ ♥➯♥ A ❧➔ Q− t ữ ỵ r K2 = K11 qt ❤ì♥✱ ♥➳✉ K1 ❧➔ ♠ët ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥ t tự K1 ỗ ợ ởt r ❝â ❣✐→ trà r✐➯♥❣ ❦❤→❝ ❦❤æ♥❣✱ ✈➔ ♥➳✉ K2 = K1−1 ✱ t❤➻ ❦❤✐ ✤â K1 0 K2 ❧➔ Sn − unita Sn ▼➺♥❤ ✤➲ ✷✳✷✳✶✳ ❈❤♦ H1 H2 t ổ ữợ tr C n ✈➔ H2 = SH1 S ∗ ✈ỵ✐ S ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ❝➜♣ n✳ ❑❤✐ ✤â A1 ❧➔ H1 − ✉♥✐t❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠❛ tr➟♥ A2 := (S ∗ )−1 A1 S ∗ ❧➔ H2 ✲ ✉♥✐t❛✳ [∗] −1 ◆➳✉ A1 ❧➔ H1− ✉♥✐t❛ t❤➻ A−1 = A1 ✳ ▼➦t ❦❤→❝ tø ✭✷✳✶✳✶✮ t❛ ❝â✿ H1 A1 = ✷✵ A∗1 H1 ✳ ❉♦ ✤â✿ ∗ ∗ −1 −1 ∗ H2 A−1 = (SH1 S )((S ) A1 S ) ∗ ∗ ∗ = SH1 A−1 S = SA1 H1 S = (SA∗1 S − 1)(SH1 S ∗ ) = A∗2 H2 ❱➟② A2 ❧➔ H2− ✉♥✐t❛✳ ▼➺♥❤ ✤➲ s❛✉ s➩ ❝❤♦ t❛ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠ët ♠❛ tr➟♥ H− ✉♥✐t❛✳ ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ H− ✉♥✐t❛✳ ❑❤✐ ✤â✱ σ(A) ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ trá♥ ✤ì♥ ✈à✱ tù❝ ❧➔✿ λ0 ∈ σ(A) ❦➨♦ t❤❡♦ λ−1 ∈ σ(A)✳ ❍ì♥ ♥ú❛✱ tr♦♥❣ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥ ❝õ❛ A✱ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ0 ✈➔ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ−1 ❧➔ ❜➡♥❣ ♥❤❛✉✳ ◆❤➢❝ ❧↕✐ r➡♥❣✿ ♥➳✉ |α| = ✈➔ w = w t❤➻ →♥❤ ①↕ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ f (z) =α(z − w)/(z − w) ✭✷✳✷✳✶✵✮ →♥❤ ①↕ tø ♥❤ú♥❣ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ tr♦♥❣ z✲ ♣❤➥♥❣ ❧➯♥ ✤÷í♥❣ trá♥ ✤ì♥ ✈à tr♦♥❣ ζ ✲ ♣❤➥♥❣✱ tr♦♥❣ ✤â ζ = f (z)✳ ⑩♥❤ ①↕ ♥❣❤à❝❤ ✤↔♦ ❧➔✿ z = (wζ − wα)/(ζ − α) ✭✷✳✷✳✶✶✮ ◆➳✉ w ∈/ σ(A) t❤➻ ❤➔♠ f ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ σ(A) ✈➔ ♥➳✉ A ❧➔ H ✲ tü ❧✐➯♥ ❤ñ♣ t❤➻ t❛ ❝â ❦➳t q✉↔ U = f (A) ❧➔ H− ✉♥✐t❛✳ ❈❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ▲➜② A ❧➔ ♠❛ tr➟♥ H ✲tü ❧✐➯♥ ❤ñ♣✳ ▲➜② w ❧➔ ♠ët sè t❤✉➛♥ ↔♦ ✈ỵ✐ w ∈ / σ(A) ✈➔ α ❧➔ sè ♣❤ù❝ ❝â ♠ỉ✤✉♥ ✤ì♥ ✈à✳ ❑❤✐ ✤â✱ U =α(A − wI)(A − wI)−1 ✭✷✳✷✳✶✷✮ ❧➔ H ✲ ✉♥✐t❛ ✈➔ α ∈ / σ(U )✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ U ❧➔ H ✲ ✉♥✐t❛✱ |α| = ✈➔ α ∈ / σ(U )✱ t❤➻ ❦❤✐ ✤â ✈ỵ✐ ❜➜t ❦➻ w = w t❤➻ ♠❛ tr➟♥✿ A =(wU − wαI)(U − αI)−1 ✭✷✳✷✳✶✸✮ ❧➔ H ✲ tü ❧✐➯♥ ❤ñ♣ ✈➔ w ∈ / σ(A)✳ ❍ì♥ ♥ú❛✱ ❤❛✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✷✳✶✷✮ ✈➔ ✭✷✳✷✳✶✸✮ ❧➔ ♥❣÷đ❝ ♥❤❛✉✳ ✷✶ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ A✲ ❧➔ H ✲ tü ❧✐➯♥ ❤ñ♣ ✈➔ |α| = t❤➻ ❞➵ ❞➔♥❣ t❤➜② ✤÷đ❝✿ (A∗ − wI)H(A − wI) = (αA∗ − αwI)H(αA − αwI) ◆❤➙♥ t❤➯♠ ✈➔♦ ❜➯♥ tr→✐ ✈➔ ❜➯♥ ♣❤↔✐ ❝õ❛ ❝↔ ❤❛✐ ✈➳ t÷ì♥❣ ù♥❣ ✈ỵ✐ (A∗ −wI)−1 ✈➔ (αA − αwI)−1✳ ❚❛ ♥❤➟♥ t❤➜② r➡♥❣✿ HU −1 = U ∗H tr♦♥❣ ✤â U ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ✭✷✳✷✳✶✷✮✱ tù❝ ❧➔ U ❧➔ H ✲✉♥✐t❛✳ ❍ì♥ ♥ú❛✱ tø ✭✷✳✷✳✶✷✮✱ t❛ ❝â✿ (U − αI)(A − wI) = α(w − w)I ✭✷✳✷✳✶✹✮ ❉♦ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t w ❧➔ sè t❤✉➛♥ ↔♦ s✉② r❛ U − αI ❧➔ ❦❤↔ ♥❣❤à❝❤ ✈➔ α∈ / σ(U )✳ ❚ø ✭✷✳✷✳✶✹✮✱ s✉② r❛✿ A = wI + α(w − w)(U − αI)−1 = [w(U − αI) + α(w − w)I](U − αI)−1 = (wU − wαI)(U − αI)−1 ♥➯♥ ✭✷✳✷✳✶✸✮ ✈➔ ✭✷✳✷✳✶✷✮ ❧➔ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♥❤❛✉✳ ●✐↔ sû r➡♥❣ U ❧➔ H− ✉♥✐t❛✱ A ❧➔ H− tü ❧✐➯♥ ❤đ♣✳ ❑❤✐ ✤â✱ ❦❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ ❝õ❛ U tữỡ ự ợ tr r (U ) ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ ❝õ❛ A tữỡ ự ợ tr r õ à0 = (wλ0 − wα)(λ0 − α)−1 ✭✷✳✷✳✶✺✮ Rλ0 (U ) = Rµ0 (A) ✭✷✳✷✳✶✻✮ ❉♦ ✤â✱ ❑➳t q✉↔ ♥➔② ❝â t❤➸ s✉② r❛ trü❝ t✐➳♣ tø ✭✷✳✷✳✶✸✮✱ t✉② ♥❤✐➯♥ ✤â ❝ô♥❣ ❧➔ ❤➺ q✉↔ ❝õ❛ ♥❤✐➲✉ ❜ê ✤➲ tê♥❣ q✉→t✳ ❇ê ✤➲ ✷✳✷✳✶✳ ▲➜② S, T ❧➔ ♠❛ tr➟♥ ❝➜♣ n t❤ä❛ ♠➣♥✿ S = f (T )✱ T = g(S)✱ ✈ỵ✐ f, g ❧➔ ❝→❝ ❤➔♠ ❣✐↔✐ t➼❝❤ tr♦♥❣ ❧➙♥ (T ) (S) tữỡ ự õ ợ ♠å✐ λ0 ∈ σ(S) t❛ ❝â✿ Rλ0 (S) = Rg(λ0 ) (T ) ✷✷ ✭✷✳✷✳✶✼✮ ▲➜② g˜ ❧➔ ✤❛ t❤ù❝ ❝â t➼♥❤ ❝❤➜t✿ g(λ0) = g˜(λ0) ✈ỵ✐ λ0 ∈ σ(S)✳ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜②✿ g(S) = g˜(S)✳ ●✐↔ sû✿ pj=0 αj (λ − λ0)j ✱ ♥➯♥ ❈❤ù♥❣ ♠✐♥❤✳ p αj (S − λ0 I)j − α0 I = W (S − λ0 I), T − g(λ0 )I = g˜(S) − g˜(λ0 )I = j=0 tr♦♥❣ ✤â W = p j=1 αj (S − λ0 I)j−1 ❧➔ ♠❛ tr➟♥ ❣✐❛♦ ❤♦→♥ ✈ỵ✐ S ✳ ❑❤✐ ✤â✿ (T − g(λ0 )I)s = W s (S − λ0 I)s , ✈ỵ✐ s = 0, 1, , ♥➯♥ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ s✉② rë♥❣✱ t❛ ❝â Rλ (S) ⊆ Rg(λ ) (T )✳ ❚✉② ♥❤✐➯♥✱ t❤❛② t❤➳ t÷ì♥❣ ù♥❣ S ✱ T ❜➡♥❣ T ✱ g(λ0 )✱ t❛ ❝â ❜❛♦ ❤➔♠ t❤ù❝✿ 0 Rg(λ0 ) (T ) ⊆ Rf (g(λ0 )) (f (T )) = Rλ0 (S), ♥➯♥ ✭✷✳✷✳✶✼✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✷✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ Rλ(U ) ✈➔ Rµ(U ) ❝õ❛ ♠ët ♠❛ tr➟♥ H− ✉♥✐t❛ U ❧➔ H− trü❝ ❣✐❛♦ ✈ỵ✐ λ = µ−1 ✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ①➨t ✤➳♥ ✤➦❝ tr÷♥❣ t❤ù ❤❛✐ ❝õ❛ ♠❛ tr➟♥ H− ✉♥✐t❛✳ ❇ê ✤➲ ✷✳✷✳✷✳ ◆➳✉ U ∗AU = A✱ detA = 0✱ t❤➻ ❝â ♠ët ♠❛ tr➟♥ H ✱ ✈ỵ✐ H ∗ = H ✈➔ detH = s❛♦ ❝❤♦ U ∗ HU = H ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû H = zA + zA∗ ✈ỵ✐ z t❤ä❛ ♠➣♥ |z| = 1✳ ❑❤✐ ✤â✱ U ∗ HU = U ∗ (zA + zA∗ )U = zA + zA∗ = H ✈➔ H ∗ = H ✳ ✣➸ ❦✐➸♠ tr❛ r➡♥❣ detH = t ú ỵ H = zA + zA = zA(z −1 zI + A−1 A∗ ) ✈➔ t❛ ❝❤➾ ♣❤↔✐ ❝❤å♥ z ✭✈ỵ✐ |z| = 1✮ ♥➯♥ −z−1z = z2 / (A1A) ỵ ởt tr U ❧➔ H− ✉♥✐t❛ ✭✈ỵ✐ H t❤ä❛ ♠➣♥✿ H ∗ = H ✱ detH = 0✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ U = A−1 A∗ ✈ỵ✐ A ❧➔ ♠❛ tr➟♥ ❦❤ỉ♥❣ s✉② ❜✐➳♥✳ ✷✸ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ U = A−1A∗ t❤➻ U ∗ AU = A(A∗ )−1 AA−1 A∗ = A ✈➔ tø ❜ê ✤➲ tr➯♥✱ U ❧➔ H− ✉♥✐t❛✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû U ❧➔ H− ✉♥✐t❛ ✈➔ A = iβ(I − αU ∗)H tr♦♥❣ ✤â |α| = 1✱ α∈ / σ(U ) ✈➔ β/β = α✳ ❑❤✐ ✤â✱ AU =iβ(I − αU ∗ )HU = iβH(U − αI) =iαβH(αU − I) = −iαβH(I − αU ) = − iβH(I − αU ) = A∗ ♥➯♥ U = A−1A∗✳ ✷✳✸✳ ❚÷ì♥❣ ✤÷ì♥❣ ❯♥✐t❛ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ●✐↔ sû A1 ✈➔ A2 ❧➛♥ ❧÷đt ❧➔ H1− tü ❧✐➯♥ ❤đ♣ ✈➔ H2− tü ❧✐➯♥ ❤đ♣ ❝➜♣ n✳ ▼❛ tr➟♥ A1 ✈➔ A2 ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛ ♥➳✉ A1 = T −1 A2 T ✱ tr♦♥❣ ✤â T ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✈➔ (H1 , H2 )− ✉♥✐t❛ ✭tù❝ ❧➔ [T x, T y]H = [x, y]H ✈ỵ✐ ♠å✐ x, y ∈ C n ❤♦➦❝ H1 = T ∗H2T ✮✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ A1 ✈➔ A2 ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛ ú ỗ tr ỗ t ố ợ t ổ ữợ t U ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❝➦♣ (A, H)✱ tr♦♥❣ ✤â A tr ự tũ ỵ H ởt t ổ ữợ ổ tr C n tự ❧➔ H ∗ = H ✈➔ detH = 0✳ ❈➦♣ (A1 , H1 )✱ (A2 , H2 ) t❤✉ë❝ U ữủ tữỡ ữỡ t ợ tr ❦❤↔ ♥❣❤à❝❤ T ✱ t❛ ❝â✿ A1 = T −1 A2 T ✈➔ H1 = T ∗ H2 T ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜②✱ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛ tr➯♥ U ❝â t➼♥❤ ❝❤➜t ♣❤↔♥ ①↕✱ ✤è✐ ①ù♥❣✱ ❜➢❝ ❝➛✉✱ tù❝ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ữỡ tr U ợ tữỡ ữỡ tữỡ ự ữủ ợ tữỡ ữỡ t U ỵ ởt ợ tữỡ ữỡ t ❝➦♣ ♠❛ tr➟♥ tr♦♥❣ U ❧✐➯♥ t❤æ♥❣ ❝✉♥❣✳ ✷✹ ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (A, H) ✈➔ (B, G) tr♦♥❣ U ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛✳ ❉♦ ✤â✱ A = S −1BS ✱ H = S ∗GS ✱ ✈ỵ✐ S ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✳ ▲➜② S(t), t ∈ [0, 1] ❧➔ ♠ët ✤÷í♥❣ ❧✐➯♥ tư❝ ❝õ❛ ❝→❝ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✈ỵ✐ S(0) = I, S(1) = S ✳ ●✐↔ sû J ❧➔ ♠ët ❞↕♥❣ ❏♦r❞❛♥ ✤è✐ ✈ỵ✐ S ✈➔ ♠é✐ ❦❤✉♥❣ ❏♦r❞❛♥ Jp = λpI + K, λp = ✭tr♦♥❣ ✤â K ❧➔ ♠❛ tr➟♥ ❧ơ② ❧✐♥❤ ✈ỵ✐ ♥❤ú♥❣ sè ✶ ♥➡♠ ♣❤➼❛ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ✈➔ ❜➡♥❣ ✵ ð ♥❤ú♥❣ ✈à tr➼ ❦❤→❝✮✱ ✤➦t Jp (t) = λp (t)I + tK ✱ ✈ỵ✐ λp (t) ❧➔ ♠ët ✤÷í♥❣ ❧✐➯♥ tư❝ ❝õ❛ ♥❤ú♥❣ sè ♣❤ù❝ ❦❤→❝ ✵ s❛♦ ❝❤♦ λp(0) = 1, λp(1) = λp✳ ❚ø ✤â✱ ❧➜② J(t) ❧➔ ❦❤è✐ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ✤÷đ❝ t↕♦ ♥➯♥ tø ♥❤ú♥❣ ❦❤è✐ Jp(t) ❝ơ♥❣ ✤ó♥❣ ♥❤÷ ❝→❝❤ ♠➔ J ✤÷đ❝ t↕♦ ♥➯♥ tø ♥❤ú♥❣ ❦❤è✐ Jp✳ ❚ø sü ①➙② ❞ü♥❣ ♥➔② ❞➝♥ ✤➳♥ J(0) = I, J(1) = Jp✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ S(t) = T J(t)T −1✱ tr♦♥❣ ✤â T ❧➔ ♠❛ tr➟♥ t❤ä❛ ♠➣♥ S = T JT −1✳ ❙û ❞ư♥❣ ✤÷í♥❣ S(t) ①➙② ❞ü♥❣ ♠ët ✤÷í♥❣ ỳ (B(t), G(t)) tr ợ tữỡ ữỡ t (G, B) ❜ð✐ B(t) = S(t)−1 BS(t), G(t) = S(t)∗ GS(t), ✈ỵ✐ t ∈ [0, 1]✳ ❑❤✐ ✤â✱ (B(0), G(0)) = (B, G) ✈➔ (B(1), G(1)) = (A, H)✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✳ ●✐↔ sû (A1, H1) ✈➔ (A2, H2) ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛✳ ❑❤✐ ✤â✱ A1 ❧➔ tü ❧✐➯♥ ❤đ♣ t ố ợ t ổ ữợ H1 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A2 ❧➔ tü ❧✐➯♥ ❤ñ♣ ✭✉♥✐t❛✮ ✤è✐ ✈ỵ✐ t➼❝❤ ①→❝ ✤à♥❤ ❜ð✐ H2 ✳ ❇ê ✤➲ ✷✳✸✳✶✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ H− ❯♥✐t❛ tổ rữợ t t t t ❝→❝ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤đ♣ ❧➔ ❧✐➯♥ t❤ỉ♥❣ ❝✉♥❣ ✈➻ ✤â ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ♥➔② ❞ü❛ ✈➔♦ ♥❤➟♥ ①➨t ♥➔②✳ ●✐↔ sû U1, U2 ❧➔ ♠❛ tr➟♥ H− ✉♥✐t❛ ✈➔ ❧➜② a1, a2 ❧➔ sè ♣❤ù❝ ❝â ♠ỉ ✤✉♥ ✤ì♥ ✈à s❛♦ ❝❤♦ U1 − a1I ✈➔ U2 − a2I ❧➔ ❦❤↔ ♥❣❤à❝❤✳ ❱ỵ✐ j = 1, 2✱ t❛ ✤à♥❤ ♥❣❤➽❛ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤ñ♣ ❈❤ù♥❣ ♠✐♥❤✳ Aj = (Uj − aj I)−1 (wUj − waj I), tr♦♥❣ ✤â w ∈ C − R✳ ✣✐➲✉ ✤â ❝❤♦ t❤➜② w ∈/ σ(A1) ∪ σ(A2) ●✐↔ sû A(t)✱ t ∈ [0, 1] ❧➔ ♠ët ✤÷í♥❣ ❧✐➯♥ tư❝ ❝õ❛ ❝→❝ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤ñ♣ s❛♦ ❝❤♦ A(0) = A1, A(1) = A2✳ ▲➜② w(t), t ∈ [0, 1] ❧➔ ♠ët ✤÷í♥❣ ✷✺ ❧✐➯♥ tư❝ tr♦♥❣ C − R t❤ä❛ ♠➣♥ w(0) = w = w(1) ✈➔ w(t) ∈/ σ(A(t)) ✈ỵ✐ t ∈ [0, 1]✳ ❈❤➥♥❣ ❤↕♥✱ t❛ ❝â t❤➸ ❝❤å♥ w(t) ❜➡♥❣ ❤➡♥❣ sè w✱ trø r❛ ♥❤ú♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ♥❤ú♥❣ ✤✐➸♠ t s❛♦ ❝❤♦ w ∈ σ(A(t))✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ U (t) = (A(t) − w(t)I)(A(t) − w(t)I)−1 ✤➸ t❤✉ ✤÷đ❝ ♠ët ✤÷í♥❣ ❝õ❛ ♥❤ú♥❣ ♠❛ tr➟♥ H− ✉♥✐t❛ ♥è✐ ❣✐ú❛ a−1 U1 ✈➔ a−1 U2 ✳ ❚ø ✤â✱ U1 , U2 ❝ô♥❣ ❧➔ ❧✐➯♥ tổ ỵ ộ ợ tữỡ ữỡ ❝→❝ ♠❛ tr➟♥ H− tü ❧✐➯♥ ❤đ♣ ❧➔ ❧✐➯♥ t❤ỉ♥❣ ❝✉♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû A1 , A2 ❧➔ H− tü ❧✐➯♥ ❤ñ♣ ✈➔ A1 = U −1 A2 U tr♦♥❣ ✤â U ❧➔ H− ✉♥✐t❛✳ ❚ø ❇ê ✤➲ ✭✷✳✸✳✶✮✱ tỗ t ởt ữớ U (t) t [0, 1] ❝õ❛ ❝→❝ ♠❛ tr➟♥ H− ✉♥✐t❛ s❛♦ ❝❤♦ U (0) = U ✈➔ U (1) = I ✳ ❑❤✐ ✤â✱ ữớ A(t) = U (t)1 A2 U (t) ợ t ∈ [0, 1] ♥è✐ ❣✐ú❛ A1 ✈➔ A2 ♥➡♠ tr♦♥❣ ợ tữỡ ữỡ tr t s ✤÷❛ r❛ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ ♠❛ tr t tỹ q ữợ G ❞ò♥❣ ✤➸ ❝❤➾ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❧➺❝❤ s❛✉✿ G2 m = Im ∈ R2m×2m −Im ✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ GT = −G, G2 = −I, G−1 = −G = GT ✭✷✳✹✳✶✽✮ ▼ët ♠❛ tr➟♥ S ❝➜♣ 2m ✤÷đ❝ ❣å✐ ❧➔ ❙②♠♣❧❡❝t✐❝ ♥➳✉✿ S T G2m S = G2m ✷✻ ✭✷✳✹✳✶✾✮ ❱➼ ❞ö ✷✳✹✳✶✳ ▼ët ♠❛ tr➟♥ t❤ü❝ ❝➜♣ ❤❛✐ S ❧➔ tü❛ ✤è✐ ①ù♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ detS = 1✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ a, b, c, d ∈ R ❝â t❤➸ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝✱ S T GS = = a c b d −1 a b c d ad − bc −ad + bc ❱➻ t❤➳✱ detS = ad − bc = ❦➨♦ t❤❡♦ S T GS = G ✈➔ ♥❣÷đ❝ ❧↕✐✱ S T GS = G ❦➨♦ t❤❡♦ detS = ad − bc = 1✳ ▼➺♥❤ ✤➲ ✷✳✹✳✶✳ ◆➳✉ S ❧➔ tü❛ ✤è✐ ①ù♥❣ t❤➻ S ❦❤↔ ♥❣❤à❝❤ ✈➔ ❝→❝ ♠❛ tr➟♥ S −1 ✱ S T ✱ −S ❝ô♥❣ ❧➔ tü❛ ✤è✐ ①ù♥❣✳ ◆➳✉ S1 , S2 ∈ R2m×2m ❧➔ ❝→❝ ♠❛ tr➟♥ tü❛ ✤è✐ ①ù♥❣ t❤➻ t➼❝❤ ❝õ❛ ❝❤ó♥❣ ❝ơ♥❣ ❧➔ ♠❛ tr➟♥ tü❛ ✤è✐ ①ù♥❣✳ ỵ S R2mì2m ởt tr➟♥ tü❛ ✤è✐ ①ù♥❣✳ ❑❤✐ ✤â✱ σ(S) ❧➔ ✤è✐ ①ù♥❣ ố ợ ữớ t tỹ ữớ trỏ ỡ tù❝ ❧➔✿ −1 λ0 ∈ σ(S) =⇒ λ0 ∈ σ(S), λ−1 ∈ σ(S), λ0 ∈ σ(S) ❍ì♥ ♥ú❛✱ tr♦♥❣ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❏♦r❞❛♥ ❝õ❛ S ✱ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ −1 ❣✐→ trà r✐➯♥❣ λ0 ✈➔ ❝➜♣ ❝õ❛ ❦❤✉♥❣ ❏♦r❞❛♥ ✈ỵ✐ ♠é✐ ❣✐→ trà r✐➯♥❣✿ λ0 , λ−1 , λ0 ❧➔ ♥❤÷ ♥❤❛✉✳ ❑➳t q✉↔ ♥➔② ✤÷đ❝ s✉② r❛ tø ▼➺♥❤ ✤➲ ✭✷✳✷✳✷✮ ✈ỵ✐ S ❧➔ ♠❛ tr➟♥ t❤ü❝ ✭tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ♣❤ê ❝õ❛ S ✤è✐ ①ù♥❣ q✉❛ trö❝ t❤ü❝✮✳ ❚❛ ❝â t❤➸ sû ❞ö♥❣ ỵ tr ợ ổ ❜➜t ❜✐➳♥ t❤ü❝ ❝õ❛ ❝→❝ ♠❛ tr➟♥ tü❛ ✤è✐ ①ù♥❣ t ố rữợ t t ự RR,λ (A) = Ker(A − λI)n ⊆ Rn ✈➔ RR,µ±iv (A) = Ker(A2 − 2µA + (µ2 + v )I)n ⊆ Rn ❧➔ ❦❤æ♥❣ ❣✐❛♥ r✐➯♥❣ s✉② rë♥❣ t❤ü❝ tr tỹ A n tữỡ ự ợ ❣✐→ trà r✐➯♥❣ λ ✈➔ ❝➦♣ ❣✐→ trà r✐➯♥❣ ❧✐➯♥ ❤đ♣ µ ± iv✳ ◆➳✉ S ❧➔ tü❛ ✤è✐ ①ù♥❣✱ ✷✼ t❛ ✤à♥❤ ♥❣❤➽❛✿ RS λ (S) := RR,λ (S) RR,µ±iv (S) R (S) R R,λ RR,µ1 ±iv1 (S) R,λ−1 (S) RR,µ2 ±iv2 (S) ♥➳✉ λ = ±1 ❀ ♥➳✉ |λ| = ✈➔ λ ❝â ♣❤➛♥ ↔♦ ❞÷ì♥❣ ❀ ♥➳✉ λ ∈ R, |λ| > 1❀ ♥➳✉ λ ❝â ♣❤➛♥ ↔♦ ❞÷ì♥❣ ✈➔ |λ| > ✭tr♦♥❣ ✤â λ := µ1 + iv1 ✈➔ λ−1 := µ2 + iv2 ✮✳ ❙❛✉ ✤➙② ❧➔ sü ♣❤➙♥ t➼❝❤ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❜➜t ❜✐➳♥ ❝õ❛ ♠❛ tr➟♥ tü❛ ✤è✐ ①ù♥❣ S ✿ Rn = RS λ (S) tr♦♥❣ ✤â ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ S tr♦♥❣ t➟♣ {1} ∪ {−1} ∪ {z ∈ C : |z| = 1, Tz > 0} ∪ {z ∈ R : |z| > 1} ❚ø ✤â✱ t❛ ❝â t❤➸ ♣❤→t ❜✐➸✉ ♠ët ❦➳t q✉↔ ✈➲ sü trü❝ ❣✐❛♦ t÷ì♥❣ tü ♥❤÷ q ỵ S tỹ ✤è✐ ①ù♥❣ ✈➔ v ∈ RS λ (S), w ∈ RS λ (S)✱ tr♦♥❣ ✤â λ1 = λ2 ✱ t❤➻ v ✈➔ w ❧➔ G− trü❝ ❣✐❛♦✱ tù❝ ❧➔ (Gv, w) = ❱ỵ✐ S ❧➔ iG− ✉♥✐t❛ t❤➻ ❦➳t q✉↔ ♥➔② ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ❍➺ q✉↔ ✭✷✳✷✳✶✮✳ ✷✽ ❑➳t ❧✉➟♥ ❚÷ì♥❣ tü ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❞↕♥❣ ❏♦r❞❛♥ ❝õ❛ ❝→❝ ♠❛ tr➟♥ t❤ü❝✱ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✤➣ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ♠ët sè ❧ỵ♣ ❝→❝ ♠❛ tr➟♥ ♣❤ù❝✳ ❈ư t❤➸ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✤➣ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè tự trú tỹ ỗ tr ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ tø ✤â →♣ ❞ư♥❣ ✈➔♦ ❝→❝ ♠❛ tr➟♥ ✈✉ỉ♥❣ ♣❤ù❝✳ ✷✳ ❚r➻♥❤ ❜➔② ❝➜✉ tró❝ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ r✐➯♥❣ ✈➔ ❝➜✉ tró❝ ♣❤ê ❝õ❛ ❝õ❛ ♠ët sè ❧ỵ♣ ❝→❝ ♠❛ tr➟♥ ✈✉ỉ♥❣ ♣❤ù❝ ✤➦❝ ❜✐➺t ♥❤÷ ✳✳✳✱ ✤➙② ❧➔ ❝→❝ ❦➳t q✉↔ ♠ð rë♥❣ tø ❝→❝ ♠❛ tr➟♥ ❍❡❝♠✐t✱ ♠❛ tr➟♥ ❯♥✐t❛✳✳✳ ✷✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✵ ... (q−1) (q−1) → − − − − α , , → α (q−1) , → α , , → α (q−1) rq rq +1 rq−1 (1) − → − (1) → − → − (1) → − (1) → − (1) α (1) α , , → rq , α rq +1 , , α rq−1 , , α r2 , , α r1 s❛♦ ❝❤♦ ❤➺ ợ tữỡ ữỡ... = 1, λp(1) = λp✳ ❚ø ✤â✱ ❧➜② J(t) ❧➔ ❦❤è✐ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ✤÷đ❝ t↕♦ ♥➯♥ tø ♥❤ú♥❣ ❦❤è✐ Jp(t) ❝ơ♥❣ ✤ó♥❣ ♥❤÷ ❝→❝❤ ♠➔ J ✤÷đ❝ t↕♦ ♥➯♥ tø ♥❤ú♥❣ ❦❤è✐ Jp✳ ❚ø sü ①➙② ❞ü♥❣ ♥➔② ❞➝♥ ✤➳♥ J(0) = I, J(1) =... B(t) = S(t)−1 BS(t), G(t) = S(t)∗ GS(t), ✈ỵ✐ t ∈ [0, 1]✳ ❑❤✐ ✤â✱ (B(0), G(0)) = (B, G) ✈➔ (B(1), G(1)) = (A, H)✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✳ ●✐↔ sû (A1, H1) ✈➔ (A2, H2) ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✉♥✐t❛✳ ❑❤✐ ✤â✱ A1 ❧➔