Open Equações polinomiais e matrizes circulantes

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❊q✉❛çõ❡s P♦❧✐♥♦♠✐❛✐s ❡ ▼❛tr✐③❡s

❈✐r❝✉❧❛♥t❡s

♣♦r

P❡❞r♦ ❏❡rô♥✐♠♦ ❙✐♠õ❡s ❞❡ ❖❧✐✈❡✐r❛ ❏ú♥✐♦r

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❏✉❧❤♦ ✴ ✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡}

❆ ❉❡✉s✱ ♣❡❧❛ ✈✐❞❛ ❡ ♣♦r t♦❞♦s ♦s ❞♦♥s ❞❛❞♦s ❛ ♠✐♠✳

❆♦s ♠❡✉s ♣❛✐s✱ ❆✈❛♥✐ ❡ P❡❞r✐♥❤♦✱ ♣♦r t♦❞♦s ♦s s❛❝r✐❢í❝✐♦s ❢❡✐t♦s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❡st✉❞❛r ♥✉♠❛ ❜♦❛ ❡s❝♦❧❛ ❡ ❝❤❡❣❛ss❡ ❛té ❛q✉✐✳

❆♦s ♠❡✉s ❞♦✐s ✜❧❤♦s ❆rt❤✉r ❍❡♥❞r✐❝❦s ❡ ❆✉❣✉st♦ ❈és❛r ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❢❛❧t❛ ❞❡ ❛t❡♥çã♦ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s✳

❆ ♠✐♥❤❛ ❡s♣♦s❛ ❆♥❛ P❛✉❧❛ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ♥❛s ♠✐♥❤❛s ❢❛❧t❛s ❜❡♠ ❝♦♠♦ ♣❡❧♦ s❡✉ ❡♠♣❡♥❤♦ ❢❛♠✐❧✐❛r ♣❛r❛ q✉❡ t✉❞♦ ❝♦rr❡ss❡ ❜❡♠✳

❆♦ ♠❡✉ ❛♠✐❣♦ ▼❛r❝♦s ❆✉ré❧✐♦ ▼❡♥❞❡s ❞❡ ▼♦r❛❡s ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥s✐stê♥❝✐❛ ❡♠ t❡r ♠❡ ❡♥s✐♥❛❞♦ ♠❛t❡♠át✐❝❛ q✉❛♥❞♦ ❡✉ t✐♥❤❛ ✶✹ ❛♥♦s✳ ❆❧é♠ ❞✐ss♦ ❞❡s♣❡rt♦✉ ❝✉r✐♦s✐❞❛❞❡ ❡ s❡❞❡ ❞❡ ❛♣r❡♥❞❡r✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡♠ ♠✐♠ ❞❡♣♦s✐t❛❞❛ ❡ ❤♦r❛s ❞❡❞✐❝❛❞❛s ❛♦ ♠❡✉ ❡♥s✐♥❛♠❡♥t♦✳ ❆❣r❛❞❡ç♦✱ t❛♠❜é♠✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❥✉♥t♦s✱ ♠♦str❛♥❞♦✲♠❡✱ ❛❧é♠ ❞❛s ♣❛❧❛✈r❛s ❞❡ s❛❜❡❞♦r✐❛✱ s❡r ♣♦ssí✈❡❧ ❢❛③❡r ❛s ❞✉❛s ❝♦✐s❛s✿ ❡st✉❞❛r ❡ s❡ ❞✐✈❡rt✐r✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❛♠✐❣♦s ◆❛❝✐❜ ❡ ▲❡♥✐♠❛r✳

❆♦ ❛♠✐❣♦ ▼❛r❝❡❧♦ ❉❛♥t❛s ♣❡❧♦ ✐♥❝❡t✐✈♦ ❝♦♥st❛♥t❡ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝♦♥❝❧✉✐r ♠❡✉ ♠❡str❛❞♦✳

❆♦ ❛♠✐❣♦ ■✈♦ ❋✐❧❤♦ ♣❡❧❛s ❝♦rr❡çõ❡s ❞♦ ♣♦rt✉❣✉ês ❡♠ ❛❧❣✉♠❛s ♣❛rt❡s ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦s ❞❡♠❛✐s ❝♦❧❡❣❛s✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❝♦♠♣❛rt✐❧❤❛❞♦s✳

✏❖ ❤♦♠❡♠ ❧ú❝✐❞♦ ♠❡ ❡s♣❛♥t❛ ♠❛s ❣♦st♦ ❞❡❧❡ ♥❛ ❧ír✐❝❛ ❆ ✈❡r❞❛❞❡ ♠❡t❛❢ís✐❝❛

♠♦❞❡❧❛ ♦ ✈❡r❜♦ ❡ ❛ ❣❛r❣❛♥t❛✳ ❖ ❤♦♠❡♠ ❧ú❝✐❞♦ ✈❡r✐✜❝❛

q✉❡ ❛ ❡①✐stê♥❝✐❛ ♥ã♦ s❡ ❡st❛♥❝❛ ♣õ❡ ❛ ❜❛❜❛ ❛♦ ♣é ❞❛ ♣❧❛♥t❛ ❡✐s q✉❡ ❛ ♣❧❛♥t❛ ❢r✉t✐✜❝❛✳ ❖ ❤♦♠❡♠ ❧ú❝✐❞♦ ❝♦♠♦ q✉❡r✱ s❡❥❛ ❧á ♦♥❞❡ ❡st✐✈❡r

❡❧❡ ❡stá✱ s❡♠ ❛q✉❛r❡❧❛✳ ❙❛❜❡ q✉❡ ❛ ✈✐❞❛ é ✈✐s❝♦s❛

s❛❜❡ q✉❡ ❡♥tr❡ ❛ ♥á✉s❡❛ ❡ ❛ r♦s❛ ❢♦✐ q✉❡ ❛ ♦str❛ ❢❛③ ❛ ♣ér♦❧❛✑

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s P❡✲ ❞r✐♥❤♦ ❡ ❆✈❛♥✐✱ ❛♦s ♠❡✉s ✜❧❤♦s✱ ❆rt❤✉r ❍❡♥❞r✐❝❦s ❡ ❆✉❣✉st♦ ❈és❛r ♣♦r ❡st❛✲ r❡♠ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ❝❛s♦s ❡ ❛❝❛s♦s ❞❛ ✈✐❞❛✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛♠♦s ✈✐❛ ♠❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉n _≤4, n_∈N∗✱ ❞❡st❛❝❛♥❞♦ ✉♠❛ ♥♦✈❛ ♣❡rs♣❡❝t✐✈❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞❛s ❢ór♠✉❧❛s ❞❡

❈❛r❞❛♥♦✲❚❛rt❛❣❧✐❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡ ♦♣♦rt✉♥✐③❛ ✉♠❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ ♦❧❤❛r ♣❛r❛ q✉❡stõ❡s ❝♦♥❡①❛s✱ ✐♥❝❧✉✐♥❞♦ ❛ ❡❧✐♠✐♥❛çã♦ ❞♦ t❡r♠♦ ❞❡ ❣r❛✉ (n₋1) ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡q✉❛çõ❡s

r❡❛✐s ❝♦♠ t♦❞❛s ❛s r❛í③❡s r❡❛✐s✳ ❖ ♠ét♦❞♦ é ❜❛s❡❛❞♦ ♥❛ ❜✉s❝❛ ❞❡ ✉♠❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡ ❝✉❥♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ s❡❥❛ ✐❞ê♥t✐❝♦ ❛♦ ❞❛s r❛í③❡s q✉❡ q✉❡r❡♠♦s ❡♥❝♦♥tr❛r✳ ❊ss❛ ♠❡t♦❞♦❧♦❣✐❛ ♥♦s ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ s✐♠♣❧❡s ❡ ✉♥✐✜❝❛❞♦ ♣❛r❛ t♦❞❛s ❡q✉❛çõ❡s ❛té q✉❛rt♦ ❣r❛✉✳

P❛❧❛✈r❛s✕❝❤❛✈❡✿ ▼❛tr✐③❡s ❞❡ ♣❡r♠✉t❛çõ❡s✳ ▼❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s✳ ❊q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳

■♥ t❤✐s ✇♦r❦ ✇❡ ❞✐s❝✉ss t❤❡ ♣r♦❝❡❞✉r❡s ❢♦r s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❡q✉❛t✐♦♥s ♦❢ ❞❡❣r❡❡ n _≤

4, n_∈ N∗ ✈✐❛ ❝✐r❝✉❧❛♥t ♠❛tr✐❝❡s✱ ❤✐❣❤❧✐❣❤t✐♥❣ ❛ ♥❡✇ ♣❡rs♣❡❝t✐✈❡ t♦ ♦❜t❛✐♥ t❤❡ ❈❛r❞❛♥♦✲

❚❛rt❛❣❧✐❛ ❢♦r♠✉❧❛❡✳ ❚❤✐s ❜r✐♥❣s ✉♣ ❛ ♥❡✇ ❧♦♦❦ ♦♥ ❝♦♥♥❡❝t❡❞ s✉❜❥❡❝ts✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ t❡r♠ ♦❢ ❞❡❣r❡❡(n₋1)❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ r❡❛❧ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤

❛❧❧ r❡❛❧ r♦♦ts✳ ❚❤❡ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ s❡❛r❝❤✐♥❣ ❛ ❝✐r❝✉❧❛♥t ♠❛tr✐① ✇❤♦s❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ s❛♠❡ r♦♦ts ✇❡ ❞❡s✐r❡ t♦ ✜♥❞✳ ❚❤✐s ❛♣♣r♦❛❝❤ ♣r♦✈✐❞❡s ✉s ❛ s✐♠♣❧❡ ❛♥❞ ✉♥✐✜❡❞ ♠❡t❤♦❞ ❢♦r ❛❧❧ ❡q✉❛t✐♦♥s t❤r♦✉❣❤ ❞❡❣r❡❡ ❢♦✉r✳

❑❡②✇♦r❞s✿ P❡r♠✉t❛t✐♦♥ ♠❛tr✐❝❡s✱ ❈✐r❝✉❧❛♥t ♠❛tr✐❝❡s✱ P♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ①

✶ ▼❛tr✐③❡s ✶

✶✳✶ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❆✉t♦✈❛❧♦r❡s ❡ ❆✉t♦✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ▼❛tr✐③ ❈✐r❝✉❧❛♥t❡ ✶✹

✷✳✶ ❈♦♥❝❡✐t♦s ❡ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ▼❛tr✐③❡s ❞❡ P❡r♠✉t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ ❊q✉❛çõ❡s ✷✽

✸✳✶ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ▼ét♦❞♦ ❈✐r❝✉❧❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✶ ◗✉❛❞rát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷✳✷ ❈ú❜✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✷✳✸ ◗✉árt✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸ ❘❛í③❡s ❘❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

❆♣ê♥❞✐❝❡ ✺✹

❆ ❇✐♦❣r❛✜❛s ✺✹

❆✳✶ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❆✳✷ ◆✐❝♦❧ò ❋♦♥t❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾

■♥tr♦❞✉çã♦

❍✐stór✐❝♦

❍á ❛❧❣♦ ❢❛s❝✐♥❛♥t❡ s♦❜r❡ ♦s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❝♦♠ ❣r❛✉ n _≤ 4, n _∈ N∗✳ P♦r ✉♠ ❧❛❞♦✱ t♦❞♦s s❛❜❡♠ q✉❡✱ ❡♥q✉❛♥t♦ s♦❧✉çõ❡s ❣❡r❛✐s

✭✉s❛♥❞♦ r❛❞✐❝❛✐s✮ sã♦ ✐♠♣♦ssí✈❡✐s ♣❛r❛ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ ❝✐♥❝♦✱ ❡❧❛s tê♠ s✐❞♦ ❡♥❝♦♥tr❛❞❛s ♣❛r❛ q✉❛❞rát✐❝❛s✱ ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛s s♦❧✉çõ❡s ❝❛♥ô♥✐❝❛s ♣❛r❛ ❛ ❝ú❜✐❝❛ ❡ ❛ q✉árt✐❝❛ sã♦ ❝♦♠♣❧✐❝❛❞❛s✱ ❡ ♦s ♠ét♦❞♦s ♣❛r❡❝❡♠ ❛❞ ❤♦❝✳ ❈♦♠♦ é q✉❡ ❛❧❣✉é♠ ♣♦❞❡ ❧❡♠❜r❛r✲s❡ ❞❡❧❛s❄ ❆ss✐♠✱ é r❛③♦á✈❡❧ ♣❡♥s❛r ❡♠ ♦❜t❡r ✉♠ ♠ét♦❞♦ s✐♠♣❧❡s ✭❞❡ ❢á❝✐❧ ♠❡♠♦r✐③❛çã♦✮ ❡ ✉♥✐✜❝❛❞♦ ♣❛r❛ t♦❞❛s ❛s ❡q✉❛çõ❡s ❞❡ ❣r❛✉s ❛té q✉❛tr♦✳

❆s ❛❜♦r❞❛❣❡♥s ♣❛r❛ ❛ ✉♥✐✜❝❛çã♦ tê♠ s✐❞♦ ❛❧❣♦ tã♦ ❧♦♥❣♦ q✉❛♥t♦ ❛s ♣ró♣r✐❛s s♦❧✉çõ❡s✳ ❈❛r❞❛♥♦✱ ❡♠ 1545✱ ♣✉❜❧✐❝♦✉ s♦❧✉çõ❡s t❛♥t♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❝ú❜✐❝❛ q✉❛♥t♦ ♣❛r❛ ❡q✉❛çã♦

q✉árt✐❝❛✱ ❛tr✐❜✉✐♥❞♦ ♦ ♣r✐♠❡✐r♦ ♠ér✐t♦ ♣❛r❛ ❚❛rt❛❣❧✐❛ ❡ ♦ s❡❣✉♥❞♦ ♣❛r❛ ❋❡rr❛r✐✳ ❉❡♣♦✐s ❞❡ ✈ár✐❛s t❡♥t❛t✐✈❛s s✉❜s❡q✉❡♥t❡s ❡ s❡♠ ê①✐t♦ ❡♠ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞❡ ❣r❛✉ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ ❝✐♥❝♦✱ ▲❛❣r❛♥❣❡✱ ❡♠1770✱ ❛♣r❡s❡♥t♦✉ ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ♣❛r❛ ❡①♣❧✐❝❛r ♣♦r

q✉❡ ♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✈❡r❡♠ ❡q✉❛çõ❡s ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s sã♦ ❜❡♠ s✉❝❡❞✐❞♦s✳ ❆ ♣❛rt✐r ❞❛q✉❡❧❡ ♠♦♠❡♥t♦ ❛té ♦ ♣r❡s❡♥t❡✱ ❡s❢♦rç♦s tê♠ s✐❞♦ ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ ❝❧❛r❡❛r ❛s s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s✱ ❝♦♥✜r❛ ❯♥❣❛r ❬✶✸❪✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ✉♥✐✜❝❛❞❛ ❝♦♠ ❜❛s❡ ❡♠ ♠❛tr✐③❡s ❝✐r✲ ❝✉❧❛♥t❡s✳ ❆ ✐❞❡✐❛ é ❝♦♥str✉✐r ✉♠❛ ✏♠❛tr✐③ ❝✐r❝✉❧❛♥t❡✑ C ❝♦♠ ✉♠ ❞❛❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝✲

t❡ríst✐❝♦ t❛❧ q✉❡ s✉❛s r❛í③❡s s❡❥❛♠ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ C q✉❡ sã♦ tr✐✈✐❛❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦s

♣❛r❛ ♠❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s✳

❊st❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ♠❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s ♣r♦♣♦r❝✐♦♥❛ ✉♠ ❜❡❧íss✐♠♦ ♠ét♦❞♦ ✉♥✐✜❝❛❞♦ ♣❛r❛ ♦❜t❡r ❛s s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s✱ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛ ❢❛❝✐❧♠❡♥t❡ ❧❡♠❜r❛❞❛s✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❛ ♠♦str❛ ♦✉tr❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❝♦♥❡①ã♦ ❡♥tr❡ ❛s ♠❛tr✐③❡s ❡

❉❡s❝r✐çã♦

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s✱ ❛✉✲ t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s✱ ♦s q✉❛✐s sã♦ ♣r❡rr❡q✉✐s✐t♦s q✉❡ ❛❧✐❝❡rç❛♠ ❛ t❡♦r✐❛ s♦❜r❡ à q✉❛❧ ❞✐ss❡rt❛r❡♠♦s✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❞✐s❝♦rr❡r❡♠♦s s♦❜r❡ ❛s ♠❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡♥tr❡ ❛s q✉❛✐s ♣♦❞❡r❡♠♦s ❞❡st❛❝❛r q✉❡ ❝❛❞❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡C ♣♦ss✉✐ ❞♦✐s ♣♦❧✐♥ô♠✐♦s

♥❛t✉r❛❧♠❡♥t❡ ❛ss♦❝✐❛❞♦s ❛ ❡❧❛✿ ♦ s❡✉ r❡♣r❡s❡♥t❛♥t❡

qC(W) = C

❡ s❡✉ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦

pC(x) =❞❡t(xI₋C).

❖s ✏❝✐r❝✉❧❛♥t❡s✑ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ 1846 ♣♦r ❈❛t❛❧❛♥ ✸ ❡♠ ❬✹❪

❖s ❝✐r❝✉❧❛♥t❡s ❞❡ ♦r❞❡♠ 2 ❡ 3✭♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉s 2 ❡ 3✮ sã♦

X0 X1

X1 X0

=X02−X12 = (X0+X1)(X0−X1)

❡

X0 X1 X2

X2 X0 X1

X1 X2 X0

= X3

0 +X13+X23−3X0X1X2

= (X0+X1+X2)(X0+ωX1 +ω2X2)(X0+ω2X1+ωX2),

❝♦♠ ω =e2πi3 ✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ♥♦s ♣r❡♥❞❡r❡♠♦s ❛ ❡ssê♥❝✐❛ ❞♦ tr❛❜❛❧❤♦ ❝♦♠❡ç❛♥❞♦ ❝♦♠ ✉♠❛ s❡çã♦ ❤✐stór✐❝❛ ❛ r❡s♣❡✐t♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❣r❛✉ ♣❡q✉❡♥♦✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛r❡♠♦s

✸_{❊✉❣è♥❡ ❈❤❛r❧❡s ❈❛t❛❧❛♥ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❜❡❧❣❛✱ q✉❡ s❡ ❞✐st✐♥❣✉✐✉ ♣❡❧♦s s❡✉s ❡st✉❞♦s s♦❜r❡ ❛ t❡♦r✐❛}

❞♦s ♥ú♠❡r♦s✳

❝♦♠♦ r❡s♦❧✈❡r ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❝✐r❝✉❧❛♥t❡✱ ❝❤❡❣❛♥❞♦ às ❢ór♠✉❧❛s ❞❡ ❈❛r❞❛♥♦✲❚❛rt❛❣❧✐❛✳ ❋✐♥❛❧✐③❛r❡♠♦s ❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ✉♠❛ ❞✐s❝✉ssã♦ s♦❜r❡ q✉❛✐s ❝r✐té✲ r✐♦s ❛s ❡q✉❛çõ❡s ❝ú❜✐❝❛s ❡ q✉árt✐❝❛s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ❞❡✈❡♠ ♣♦ss✉✐r ♣❛r❛ q✉❡ t♦❞❛s ❛s r❛í③❡s s❡❥❛♠ r❡❛✐s✳

▼❛tr✐③❡s

❆♦ tr❛❜❛❧❤❛r ❝♦♠ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ s♦♠❡♥t❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ♣♦s✐çõ❡s sã♦ ✐♠♣♦rt❛♥t❡s✳ ❚❛♠❜é♠✱ ❛♦ r❡❞✉③✐r ♦ s✐st❡♠❛ à ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛✱ é ❡ss❡♥❝✐❛❧ ♠❛♥t❡r ❛s ❡q✉❛çõ❡s ❝✉✐❞❛❞♦s❛♠❡♥t❡ ❛❧✐♥❤❛❞❛s✳ ❆ss✐♠ ❡ss❡s ❝♦❡✜❝✐❡♥t❡s ♣♦❞❡♠ s❡r ❡✜❝✐❡♥t❡♠❡♥t❡ ❛rr✉♠❛❞♦s ♥✉♠❛ ❞✐s♣♦s✐çã♦ r❡t❛♥❣✉❧❛r ❝❤❛♠❛❞❛ ♠❛tr✐③✳

❆ ♠❡♥♦s q✉❡ s❡❥❛ ❞❡❝❧❛r❛❞♦ ♦ ❝♦♥trár✐♦✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❡♠ ♥♦ss❛s ♠❛tr✐③❡s ♣❡rt❡♥❝❡♠ ❛ ❛❧❣✉♠ ❝♦r♣♦ K✱ ❛r❜✐trár✐♦ ♠❛s ✜①♦✳ ✭❈♦♥✜r❛ ❛♣ê♥❞✐❝❡ ❇✳✻✮ ❊ss❡s ❡❧❡♠❡♥t♦s

❞❡Ksã♦ ❝❤❛♠❛❞♦s ❡s❝❛❧❛r❡s✳ ◆❛❞❛ ❡ss❡♥❝✐❛❧ é ♣❡r❞✐❞♦ s❡ ♦ ❧❡✐t♦r s✉♣õ❡ q✉❡Ké ♦ ❝♦r♣♦

r❡❛❧ R ♦✉ ♦ ❝♦r♣♦ ❝♦♠♣❧❡①♦ C✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ❡♠ ♠❛✐s ❞❡t❛❧❤❡s ♣♦❞❡ ❝♦♥s✉❧t❛r

❍♦✛♠❛♥✱ ▲✐♣s❝❤✉t③ ❡✴♦✉ ❙✐❧✈❛ ❬✽✱ ✶✶✱ ✶✷❪✳

✶✳✶ ▼❛tr✐③❡s

❙❡❥❛ K ✉♠ ❝♦r♣♦ ❛r❜✐trár✐♦✳ ❯♠❛ ❞✐s♣♦s✐çã♦ r❡t❛♥❣✉❧❛r ❞❛ ❢♦r♠❛



       

a11 a12 a13 · · · a1n a21 a22 a23 · · · a2n a31 a32 a33 · · · a3n

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

am1 am2 am3 · · · amn 

       

♦♥❞❡ ♦s aij sã♦ ❡❧❡♠❡♥t♦s ❞❡K✱ é ❝❤❛♠❛❞❛ ♠❛tr✐③ s♦❜r❡ K♦✉✱ s✐♠♣❧❡s♠❡♥t❡ ♠❛tr✐③✱ s❡ K ❡stá ✐♠♣❧í❝✐t♦✳

✶✳✶✳ ▼❆❚❘■❩❊❙

❆sn✲✉♣❧❛s ❤♦r✐③♦♥t❛✐s sã♦ ❛s ❧✐♥❤❛s ❞❛ ♠❛tr✐③ ❡ ❛sm✲✉♣❧❛s ✈❡rt✐❝❛✐s sã♦ ❛s ❝♦❧✉♥❛s✳

◆♦t❡ q✉❡ ♦ ❡❧❡♠❡♥t♦ aij✱ ❝❤❛♠❛❞♦ t❡r♠♦ ❣❡r❛❧ q✉❡ ❛♣❛r❡❝❡ ♥❛ i✲és✐♠❛ ❧✐♥❤❛ ❡ j✲és✐♠❛

❝♦❧✉♥❛✳ ❆ ♠❛tr✐③ ❝♦♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s é ❝❤❛♠❛❞❛ ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ m ♣♦r n

(m_×n)✱ ❞❡s❝r❡✈❡♥❞♦✱ ❞❡ss❡ ♠♦❞♦✱ s✉❛ ❢♦r♠❛✳

❯♠❛ ♠❛tr✐③ ❝♦♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s é ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③ q✉❛✲ ❞r❛❞❛✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❝♦♠ n ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s é ❞❡ ♦r❞❡♠ n✳ ❆

❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A = (aij) ❞❡ ♦r❞❡♠ n✱ ❝♦rr❡s♣♦♥❞❡ ❛♦s ❡❧❡✲

♠❡♥t♦s ❞❡A q✉❛♥❞♦ i=j✳

❙❡❥❛♠ A = (aij)m×n ❡ B = (bij)m×n ❞✉❛s ♠❛tr✐③❡s✳ ❉✐③❡♠♦s q✉❡ A é ✐❣✉❛❧ ❛ B✱ ❡♠

sí♠❜♦❧♦sA=B✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

aij =bij, 1_≤i_≤m ❡ 1_≤j _≤n.

❉❡♥♦♠✐♥❛✲s❡ ♠❛tr✐③ ❞✐❛❣♦♥❛❧✱ ❛ t♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❡♠ q✉❡ aij = 0✱ ♣❛r❛ i₆= j✱

✐st♦ é✱ ♦s ❡❧❡♠❡♥t♦s q✉❡ ♥ã♦ ❡stã♦ ♥❛ ✏❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✑ sã♦ ♥✉❧♦s✳

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ é ❞❡♥♦♠✐♥❛❞❛ ✐❞❡♥t✐❞❛❞❡ s❡aij = 1✱ ♣❛r❛i=j✱ ❡ aij = 0✱ ♣❛r❛

i₆=j✳ ❆ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ✭q✉❡ ✐♥❞✐❝❛r❡♠♦s ♣♦r I✮ é ✉♠ ✐♠♣♦rt❛♥t❡ ❡①❡♠♣❧♦ ❞❡ ♠❛tr✐③

❞✐❛❣♦♥❛❧ q✉❡ ♠❡r❡❝❡ ❞❡st❛q✉❡ ♥❛ t❡♦r✐❛ ❞❛s ♠❛tr✐③❡s ♣♦r ❡①❡r❝❡r ♦ ♣❛♣❡❧ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ♦✉ s❡❥❛✱

A_·I =I_·A=A,

♣❛r❛ t♦❞❛ ♠❛tr✐③ A✳

❆ s♦♠❛ ❞❡ ♠❛tr✐③❡sAm×n= (aij)❡Bm×n = (bij)é ✉♠❛ ♠❛tr✐③m×n✱ q✉❡ ❞❡♥♦t❛r❡♠♦s A+B✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ s♦♠❛s ❞♦s ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✱ ✐st♦ é✱ ❡❧❡♠❡♥t♦s ❞❡

♠❡s♠❛ ♣♦s✐çã♦✱ ❞❡A ❡ B✳ ■st♦ é✱

A+B = (aij +bij)m×n.

◆♦t❡ q✉❡✱ ❞❛ ❢♦r♠❛ ❝♦♠♦ ❢♦✐ ❞❡✜♥✐❞❛✱ ❛ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s ♣♦ss✉✐ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛ ❛❞✐çã♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳

❆ ♦♣❡r❛çã♦ q✉❡ ❞❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r é à ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ♠❛tr✐③ ♣♦r ✉♠ ❡s❝❛❧❛r ❝❤❛♠❛❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✳

❙❡❥❛♠ A= (aij)m×n ❡k ∈K✳ ❉❡✜♥✐♠♦s ✉♠❛ ♥♦✈❛ ♠❛tr✐③ k_·A= (kaij)m×n.

❉❛❞❛ ✉♠❛ ♠❛tr✐③A = (aij)m×n✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ♦✉tr❛ ♠❛tr✐③AT = (bij)n×m✱ ❝✉❥❛s

❧✐♥❤❛s sã♦ ❛s ❝♦❧✉♥❛s ❞❡A✱ ✐st♦ é✱bij =aji✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡AT _{é tr❛♥s♣♦st❛ ❞❡} A✳

❯♠❛ ♠❛tr✐③ é ❞❡♥♦♠✐♥❛❞❛ s✐♠étr✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ ❢♦r ✐❣✉❛❧ à s✉❛ tr❛♥s♣♦st❛✱ ✐st♦ é✱ A=AT_✳

❉❛❞❛s ❛s ♠❛tr✐③❡s A = (aij)m×n ❡ B = (brs)n×p✱ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s

♠❛tr✐③❡s A ♣♦r B✱ ♦✉ s❡❥❛✱ AB = (cuv)m×p✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ cuv =

n

X

k=1

aukbkv =au1b1v +· · ·+aunbnv.

❖❜s❡r✈❛çã♦ ✶✳✶✳ ❊♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ♠❛tr✐③❡s ♣♦❞❡♠♦s ♥♦t❛r q✉❡✿

✶✳ ❙ó ♣♦❞❡♠♦s ❡❢❡t✉❛r ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ♠❛tr✐③❡sAm×n ❡Bl×p s❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s

❞❛ ♣r✐♠❡✐r❛ ❢♦r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛✱ ✐st♦ é✱ n =l✳ ❆❧é♠ ❞✐ss♦✱ ❛

♠❛tr✐③ C =AB s❡rá ❞❡ ♦r❞❡♠ m_×p✳

✷✳ ❖ ❡❧❡♠❡♥t♦ cij (✐✲és✐♠❛ ❧✐♥❤❛ ❡ ❥✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ♣r♦❞✉t♦) é ♦❜t✐❞♦✱ ♠✉❧t✐✲

♣❧✐❝❛♥❞♦ ♦s ❡❧❡♠❡♥t♦s ❞❛ ✐✲és✐♠❛ ❧✐♥❤❛ ❞❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❝♦rr❡s✲ ♣♦♥❞❡♥t❡s ❞❛ ❥✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③✱ ❡ s♦♠❛♥❞♦ ❡st❡s ♣r♦❞✉t♦s✳

❙❡❥❛♠ A ❡B ♠❛tr✐③❡s q✉❛✐sq✉❡r✳ ❊♠ ❣❡r❛❧✱

AB ₆=BA

✭♣♦❞❡♥❞♦ ♠❡s♠♦ ✉♠ ❞♦s ♠❡♠❜r♦s ❡stá ❞❡✜♥✐❞♦ ♦✉ ♥ã♦✮✳ ◆♦t❡✱ ❛✐♥❞❛✱ q✉❡ ♣♦❞❡♠♦s t❡r

AB = 0✱ s❡♠ q✉❡A = 0 ♦✉B = 0✳

❉✐③✲s❡ q✉❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A é ✐♥✈❡rtí✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ B t❛❧ q✉❡

A_·B =B_·A =I.

✶✳✶✳ ▼❆❚❘■❩❊❙

❆ ♠❛tr✐③B é ú♥✐❝❛✱ ♣♦✐s ❛s ✐❣✉❛❧❞❛❞❡s

AB1 =B1A =I ❡ AB2 =B2A=I

✐♠♣❧✐❝❛♠ q✉❡

B1 =B1I =B1(AB2) = (B1A)B2 =IB2 =B2.

❉❡♥♦♠✐♥❛r❡♠♦s t❛❧ ♠❛tr✐③ B ❛ ✐♥✈❡rs❛ ❞❡ A ❡ ❞❡♥♦t❛♠♦s ♣♦r A−1_{✳ ❖❜s❡r✈❡ q✉❡ ❛}

r❡❧❛çã♦ ❛❝✐♠❛ é s✐♠étr✐❝❛❀ ✐st♦ é✱ s❡B é ❛ ✐♥✈❡rs❛ ❞❡ A✱ ❡♥tã♦ A é à ✐♥✈❡rs❛ ❞❡ B✳

❊①❡♠♣❧♦ ✶✳✶✳ ❈❛❧❝✉❧❡♠♦s✱ ❛❣♦r❛✱ ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ♠❛tr✐③ ❣❡♥ér✐❝❛✱ 2_×2✱

A= a b

c d

!

.

Pr♦❝✉r❛♠♦s ❡s❝❛❧❛r❡s x, y, z, w t❛✐s q✉❡

a b c d ! x y z w !

= 1 0 0 1

!

♦✉ ax+bz ay+bw

cx+dz cy+dw

!

= 1 0 0 1

!

♦ q✉❡ s❡ r❡❞✉③ ❛ r❡s♦❧✈❡r ♦s s❡❣✉✐♥t❡s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✿

(

ax + bz = 1

cx + dz = 0 ❡

(

ay + bw = 0

cy + dw = 1

❙❡ ad₋bc₆= 0✱ ❡♥tã♦

x= d

ad₋bc, y =

−b ad₋bc z = −c

ad₋bc, w= a ad₋bc

❡ t❛✐s s♦❧✉çõ❡s sã♦ ú♥✐❝❛s✳ ◆♦t❡ q✉❡✱ ❢❛③❡♥❞♦ |A_|=ad₋bc✱ ♦❜t❡♠♦s A−1 =

d

|A| −

b

|A|

− c

|A|

a

|A|

!

= 1

|A_|

d ₋b

−c a

!

.

✶✳✷ ❉❡t❡r♠✐♥❛♥t❡s

❏á ❡♠ 250❛✳❈✳ ❤❛✈✐❛ ❡①❡♠♣❧♦s ❞❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❛tr❛✈és ❞❡ ♠❛tr✐✲

③❡s✱ ♥♦ ❧✐✈r♦ ❝❤✐♥ês ◆♦✈❡ ❈❛♣ít✉❧♦s s♦❜r❡ ❛ ❆rt❡ ▼❛t❡♠át✐❝❛✱ ❝✉❥♦ ❛✉t♦r é ❞❡s❝♦♥❤❡❝✐❞♦✳ ❚❛♠❜é♠ ❛❧❣✉♠❛s ♥♦çõ❡s ❧✐❣❛❞❛s ❛ ❞❡t❡r♠✐♥❛♥t❡s✱ ♦ ❛ss✉♥t♦ q✉❡ s❡rá ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ♥❡st❡ s❡çã♦✱ ❥á ❡r❛♠ ❝♦♥❤❡❝✐❞❛s ♥❛ ❈❤✐♥❛ ❛♥t✐❣❛✳

▼❛s✱ s❡ ♣♦r ✉♠ ❧❛❞♦ ❥á s❡ ✉t✐❧✐③❛✈❛ ❛ ♥♦çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ♥♦ ♠✉♥❞♦ ❖r✐❡♥t❛❧ ❤á t❛♥t♦ t❡♠♣♦✱ ♥♦ ❖❝✐❞❡♥t❡ ❡st❡ ❛ss✉♥t♦ ❝♦♠❡ç♦✉ ❛ s❡r tr❛t❛❞♦ ❡s♣♦r❛❞✐❝❛♠❡♥t❡ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳❱■■✳ ◆❡st❛ é♣♦❝❛ s✉r❣❡♠ tr❛❜❛❧❤♦s ❞❡ ●✳ ❲✳ ▲❡✐❜♥✐③ ✭1646₋ 1716✮✱ ❡ ●✳

❈r❛♠❡r ✭1704₋ 1752✮ q✉❡ ❞❡s❡♥✈♦❧✈❡ ✉♠ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❛tr❛✈és ❞❡

❞❡t❡r♠✐♥❛♥t❡s✱ ❝♦♥❤❡❝✐❞❛ ♣♦r ✏❘❡❣r❛ ❞❡ ❈r❛♠❡r✑ ❡ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠1750✭♣r♦✈❛✈❡❧♠❡♥t❡

❥á ❝♦♥❤❡❝✐❞♦ ♣♦r ❈✳ ▼❛❝❧❛✉r✐♥ ✭1698₋1746✮✮ ❡♠1729✱ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s s✐♠étr✐❝♦s ❞❡

❏✳ ▲✳ ▲❛❣r❛♥❣❡ ✭1736₋1813✮✳

❙ó ♥♦ sé❝✉❧♦ ❳■❳ é q✉❡ ♦s ❞❡t❡r♠✐♥❛♥t❡s ♣❛ss❛r❛♠ ❛ s❡r ❡st✉❞❛❞♦s ♠❛✐s s✐st❡♠❛t✐❝❛✲ ♠❡♥t❡✱ ❛ ❝♦♠❡ç❛r ♣❡❧♦ ❧♦♥❣♦ tr❛t❛❞♦ ❞❡ ❆✳ ▲✳ ❈❛✉❝❤② ✭1789₋1857✮ ❡♠ 1812✱ t❡♥❞♦ s✐❞♦

r❡❛❧✐③❛❞♦s✱ ❡♠ s❡❣✉✐❞❛✱ tr❛❜❛❧❤♦s ❞❡ ❈✳ ●✳ ❏❛❝♦❜✐ ✭1804₋1851✮✳

❆ ♣❛rt✐r ❞❡ ❡♥tã♦✱ ♦ ✉s♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❞✐❢✉♥❞✐✉✲s❡ ♠✉✐t♦ ❡ ❡st❡ ❝♦♥❝❡✐t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♠♦str♦✉✲s❡ ❡①tr❡♠❛♠❡♥t❡ út✐❧ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ♠✉✐t❛s s✐t✉❛çõ❡s✱ ❝♦♠♦ ❛ ❞❡ s❛❜❡r s❡ ✉♠❛ ♠❛tr✐③ é ✐♥✈❡rsí✈❡❧ ❡ s❡ ✉♠ s✐st❡♠❛ ❛❞♠✐t❡ ♦✉ ♥ã♦ s♦❧✉çã♦✳

❆ t♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A s♦❜r❡ ✉♠ ❝♦r♣♦K ❡stá ❛ss♦❝✐❛❞♦ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡K

❝❤❛♠❛❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ A✱ ✉s✉❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ♣♦r

❞❡t(A) ♦✉ _|A_|.

❆ ❢✉♥çã♦ ❞❡t❡r♠✐♥❛♥t❡ ❢♦✐ ❞❡s❝♦❜❡rt❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❡ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s t❛♠❜é♠ ❢✉♥❝✐♦♥❛♠ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ ♣❡rt❡♥❝❡♠ ❛ ✉♠ ❛♥❡❧ ✭❝♦♥✜r❛ ❛♣ê♥❞✐❝❡ ❇✳✶✮✳

❙❡❥❛♠ A❡B ❝♦♥❥✉♥t♦s ❛r❜✐trár✐♦s✳ ❙✉♣♦♥❤❛♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛a _∈A✱ ❡stá ❛ss♦❝✐❛❞♦

❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡f(a)_∈B ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛f✳ ❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ f é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ♦✉ tr❛♥s❢♦r♠❛çã♦ ❞❡ A ❡♠ B ❡ ❡s❝r✐t❛

f :A _−→B ♦✉ A_−→f B.

✶✳✷✳ ❉❊❚❊❘▼■◆❆◆❚❊❙

❊s❝r❡✈❡♠♦s f(a)✱ ❧❡✐❛ ✏f ❞❡ a✑✱ ♣❛r❛ ♦ ❡❧❡♠❡♥t♦ ❞❡ B q✉❡ f ❛ss♦❝✐❛ ❛ a _∈ A❀ f(a) é

❝❤❛♠❛❞♦ ♦ ✈❛❧♦r ❞❡ f ❡♠ a ♦✉ ❛ ✐♠❛❣❡♠ ❞❡a s♦❜ f✳

❯♠❛ tr❛♥s❢♦r♠❛çã♦ f : A _−→ B é ✐♥❥❡t♦r❛✱ s❡ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ A ♣r♦❞✉③❡♠

✐♠❛❣❡♥s ❞✐st✐♥t❛s ❡♠B✱ ✐st♦ é✱

x₆=x′ _⇒f(a)₆=f(a′)

♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

f(a) = f(a′)_⇒a =a′.

❯♠❛ ❢✉♥çã♦ f : A _−→ B é s♦❜r❡❥❡t♦r❛✱ s❡ ❝❛❞❛ y _∈ B é ✐♠❛❣❡♠ ❞❡ ❛♦ ♠❡♥♦s ✉♠ x_∈A✳ ❆❧é♠ ❞✐ss♦✱ ✉♠❛ ❢✉♥çã♦ q✉❡ é ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✱ ❞❡♥♦♠✐♥❛✲s❡ ❜✐❥❡t♦r❛✳

❯♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛ σ ❞♦ ❝♦♥❥✉♥t♦ _{1,2,3, . . . , n_} s♦❜r❡ s✐ ♠❡s♠♦ é ❝❤❛♠❛❞❛ ✉♠❛

♣❡r♠✉t❛çã♦✳ ❉❡♥♦t❛♠♦s σ ♣♦r

σ = 1 2 . . . n

j1 j2 . . . jn !

♦✉ σ = (j1j2· · ·jn), ❡♠ q✉❡ji =σ(i).

❖❜s❡r✈❡ q✉❡ ❝♦♠♦ σ é ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✱ ❛ s❡q✉ê♥❝✐❛ (j1, j2, . . . , jn) é s✐♠♣❧❡s♠❡♥t❡

✉♠ r❡❛rr❛♥❥♦ ❞♦s ♥ú♠❡r♦s 1,2, . . . , n✳ ❖ ♥ú♠❡r♦ ❞❡ t❛✐s ♣❡r♠✉t❛çõ❡s é ♥✦ ❡ ♦ ❝♦♥❥✉♥t♦

❞❡ t♦❞❛s ❛s ♣❡r♠✉t❛çõ❡s é ❞❡♥♦t❛❞❛ ♣♦r Sn✳ ❚❛♠❜é♠ ♦❜s❡r✈❛♠♦s q✉❡ s❡ σ _∈ Sn✱ ❡♥tã♦

❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rs❛ σ−1 _{∈ Sn❀ ❡✱ s❡ σ, τ ∈ Sn✱ ❡♥tã♦ ❛ tr❛♥s❢♦r♠❛çã♦ ❝♦♠♣♦st❛}

σ_◦τ _∈Sn✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ tr❛♥s❢♦r♠❛çã♦ ✐❞❡♥t✐❞❛❞❡

ǫ=σ_◦σ−1 =σ−1_◦σ

♣❡rt❡♥❝❡ ❛ Sn✳

❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ♣❡r♠✉t❛çã♦ ❛r❜✐trár✐❛ σ ❡♠

Sn :σ= (j1j2· · ·jn).

❉✐r❡♠♦s q✉❡σ é ♣❛r ♦✉ í♠♣❛r✱ ❝♦♥❢♦r♠❡ ❡①✐st❛ ✉♠ ♥ú♠❡r♦ ♣❛r ♦✉ í♠♣❛r ❞❡ ♣❛r❡s (i, k)✱

♣❛r❛ ♦s q✉❛✐s

i > j, ♠❛s i ♣r❡❝❡❞❡ k ❡♠ σ.

❊♥tã♦✱ ❞❡✜♥✐♠♦s ♦ s✐♥❛❧ ♦✉ ♣❛r✐❞❛❞❡ ❞❡σ✱ ❡s❝r✐t♦ s❣♥σ✱ ♣♦r

s❣♥σ=

(

1, s❡ σ é ♣❛r

−1, s❡ σ é í♠♣❛r✳

❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A = (aij) ❞❡ ♦r❞❡♠ n✱ é ❛ s♦♠❛ ❡❢❡t✉❛❞❛

s♦❜r❡ t♦❞❛s ❛s ♣❡r♠✉t❛çõ❡s σ = (j1j2· · ·jn) ❡♠ Sn✱ ✐st♦ é✱

❞❡t(A) = X

σ∈Sn

(s❣♥σ)

n

Y

i=1

aiσ(i)

❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A = (aij) ❞❡ ♦r❞❡♠n✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ♣♦rMij

❛ s✉❜♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ (n ₋1) ❞❡ A✱ ♦❜t✐❞❛ s✉♣r✐♠✐♥❞♦✲s❡ s✉❛ i✲és✐♠❛ ❧✐♥❤❛

❡ j✲és✐♠❛ ❝♦❧✉♥❛✳ ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡t(Mij) é ❞❡♥♦♠✐♥❛❞♦ ♦ ♠❡♥♦r ❝♦♠♣❧❡♠❡♥t❛r ❞♦

❡❧❡♠❡♥t♦ aij ❞❡A ❡ ❞❡✜♥✐♠♦s ♦ ❝♦❢❛t♦r ❞❡ aij✱ ❞❡♥♦t❛❞♦ ♣♦rAij✱ ❝♦♠♦

Aij = (₋1)i+j _·❞❡t(Mij).

❈♦♠ ❡st❡s ❝♦❢❛t♦r❡s ♣♦❞❡♠♦s ❢♦r♠❛r ✉♠❛ ♥♦✈❛ ♠❛tr✐③_A¯_{✱ ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③ ❞♦s ❝♦❢❛t♦r❡s}

❞❡ A✳

¯

A = [Aij].

❆ ♣❛rt✐r ❞❡st❛s ❝♦♥s✐❞❡r❛çõ❡s ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A= (aij)♣♦❞❡

s❡r ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱

❞❡t(A) =

n

X

j=1

aijAij ♦✉ ❞❡t(A) =

n

X

i=1

aijAij.

❆s ❢ór♠✉❧❛s s✉♣r❛❝✐t❛❞❛s✱ ❞❡♥♦♠✐♥❛♠✲s❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ▲❛♣❧❛❝❡✱ s♦❜r❡ ✉♠❛ ♠❛✲ tr✐③ q✉❛❞r❛❞❛ A= (aij) ❞❡ ♦r❞❡♠ n✱ s❡❣✉♥❞♦ ❛ i✲és✐♠❛ ❧✐♥❤❛ ❡ j✲és✐♠❛ ❝♦❧✉♥❛✱ r❡s♣❡❝t✐✲

✈❛♠❡♥t❡✱ ♦❢❡r❡❝❡♠ ✉♠ ♠ét♦❞♦ ❞❡ s✐♠♣❧✐✜❝❛r ♦ ❝á❧❝✉❧♦ ❞♦ ❞❡t(A)✳

❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛A= (aij)❞❡ ♦r❞❡♠ns♦❜r❡ ✉♠ ❝♦r♣♦K✳ ❉❡♥♦♠✐♥❛✲

s❡ ❛❞❥✉♥t❛ ❞❡A✱ ❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❞♦s ❝♦❢❛t♦r❡s ❞♦s ❡❧❡♠❡♥t♦saij ❞❡A✱ r❡♣r❡s❡♥t❛❞❛

✶✳✸✳ ❆❯❚❖❱❆▲❖❘❊❙ ❊ ❆❯❚❖❱❊❚❖❘❊❙

♣♦r ❛❞❥A✱ à ♠❛tr✐③

❛❞❥A=



    

A11 A21 . . . An1

A12 A22 . . . An2

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

A1n A2n . . . Ann



    

= ¯AT

❚❡♦r❡♠❛ ✶✳✶✳ P❛r❛ q✉❛❧q✉❡r ♠❛tr✐③ q✉❛❞r❛❞❛ A✱

A_·(❛❞❥A) = (❛❞❥A)_·A=❞❡t(A)_·I.

❆ss✐♠✱ s❡ ❞❡t(A)₆= 0✱ ❡♥tã♦

A−1 = 1

❞❡t(A)·(❛❞❥A).

✶✳✸ ❆✉t♦✈❛❧♦r❡s ❡ ❆✉t♦✈❡t♦r❡s

❆✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s sã♦ ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ❞❡ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❡♠ ár❡❛s ❞✐✈❡rs✐✜❝❛❞❛s ❝♦♠♦ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♠❛❣❡♥s✱ ❛♥á✲ ❧✐s❡ ❞❡ ✈✐❜r❛çõ❡s✱ ♠❡❝â♥✐❝❛ ❞♦s só❧✐❞♦s✱ ❡st❛tíst✐❝❛ ❡t❝✳

❙❡❥❛♠V ❡U ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ ♦ ♠❡s♠♦ ❝♦r♣♦K✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦_{F :V −→}

U ❞❡♥♦♠✐♥❛✲s❡ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r (♦✉ ❤♦♠♦♠♦r✜s♠♦)✶ s❡ s❛t✐s❢❛③ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s

s❡❣✉✐♥t❡s✿

✶✳ P❛r❛ q✉❛✐sq✉❡r v, w_∈V, F(v+w) =F(v) +F(w)✳

✷✳ P❛r❛ q✉❛❧q✉❡r k _∈K ❡ _{v ∈V, F(kv) = kF(v)}✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ F :V _−→U é ❧✐♥❡❛r s❡ ✏♣r❡s❡r✈❛✑ ❛s ❞✉❛s ♦♣❡r❛çõ❡s ❜ás✐❝❛s ❞❡ ✉♠

❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ✐st♦ é✱ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✳

❙✉❜st✐t✉✐♥❞♦ k = 0 ♥❛ ❝♦♥❞✐çã♦ ✭2✮ ❛❝✐♠❛ ♦❜t❡♠♦s F(0) = 0✳ ■st♦ é✱ t♦❞❛ tr❛♥s❢♦r✲

♠❛çã♦ ❧✐♥❡❛r ❧❡✈❛ ♦ ✈❡t♦r ♥✉❧♦ ♥♦ ✈❡t♦r ♥✉❧♦✳

P❛r❛ q✉❛✐sq✉❡r a, b_∈ K ❡ ✈❡t♦r❡s _{v, w ∈ V} ♦❜t❡♠♦s✱ ❛♣❧✐❝❛♥❞♦ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s ❞❡

❧✐♥❡❛r✐❞❛❞❡✱ q✉❡

F(av+bw) = F(av) +F(bw) = aF(v) +bF(w). ✶_{✭✈❡r ❛♣ê♥❞✐❝❡ ❇✳✽✮}

❙❡❥❛ A✉♠❛ ♠❛tr✐③ m_×n s♦❜r❡ ✉♠ ❝♦r♣♦K✳ ❊♥tã♦_A ❞❡t❡r♠✐♥❛ ✉♠❛ ú♥✐❝❛ tr❛♥s❢♦r✲

♠❛çã♦ T : Kn _{−→ K}m _{♣❡❧❛ ❛ss♦❝✐❛çã♦ v 7−→ Av✳ ❆✜r♠❛♠♦s q✉❡ T é ❧✐♥❡❛r✱ ♣♦✐s ♣❡❧❛s}

♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠❛tr✐③❡s✱ t❡♠♦s

T(v+w) = A(v+w) =Av+Aw=T(v) +T(w)

❡

T(kv) =A(kv) = k(Av) = kT(v),

♦♥❞❡✱ v, w_∈Kn _{❡ k∈K✳}

❙❡❥❛ F :V _−→U ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❆ ✐♠❛❣❡♠ ❞❡ F✱ ❡♠ sí♠❜♦❧♦s

■♠F,

é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✐♠❛❣❡♥s ❞❡ U✿

■♠F =_{u_∈U :F(v) = u, ♣❛r❛ ❛❧❣✉♠ v _∈V_}.

❖ ♥ú❝❧❡♦ ❞❡ F✱ ❡♠ sí♠❜♦❧♦s ❑❡rF✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❡♠V✱ q✉❡ sã♦ tr❛♥s✲

❢♦r♠❛❞♦s ❡♠ 0_∈U✿

❑❡rF =_{v _∈V :F(v) = 0_}.

❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r F : V _−→ U é s✐♥❣✉❧❛r s❡ ❡①✐st✐r ✉♠ v _∈ V✱ ❝♦♠ v ₆= 0✱

t❛❧ q✉❡ F(v) = 0✳ ❆ss✐♠✱ F :V _−→U é ♥ã♦✲s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❑❡rF =_{0_} s❡✱ ❡

s♦♠❡♥t❡ s❡✱ F é ✐♥❥❡t♦r❛✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❜✐❥❡t✐✈❛ é ❞❡♥♦♠✐♥❛❞❛ ✐s♦♠♦r✜s♠♦✳

❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ✉♠ ❝♦r♣♦ K✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ tr❛♥s✲

❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s T :V _−→V✱ ✐st♦ é✱ ❞❡V ♥❡❧❡ ♠❡s♠♦✳ ❊❧❛s sã♦ t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛s

♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s✳

❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦❧✐♥ô♠✐♦ f s♦❜r❡ ✉♠ ❝♦r♣♦K✿

f(x) =anxn₊

· · ·+a1x+a0.

❙❡A é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ s♦❜r❡K✱ ❡♥tã♦ ❞❡✜♥✐♠♦s ❛ ♠❛tr✐③ ♣♦❧✐♥♦♠✐❛❧ s♦❜r❡ K❝♦♠♦

f(A) =anAn₊

· · ·+a1A+a0I,

✶✳✸✳ ❆❯❚❖❱❆▲❖❘❊❙ ❊ ❆❯❚❖❱❊❚❖❘❊❙

❊♠ ♣❛rt✐❝✉❧❛r✱ ❞✐r❡♠♦s q✉❡A é ✉♠❛ r❛✐③ ♦✉ ③❡r♦ ❞♦ ♣♦❧✐♥ô♠✐♦f s❡f(A) = 0✳

❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ T :V _−→V é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r s♦❜r❡ V✳ ❙❡ f(x) = anxn+_{· · ·}+a1x+a0,

❡♥tã♦ ❞❡✜♥✐♠♦sf(T) ❞♦ ♠❡s♠♦ ♠♦❞♦ ❝♦♠♦ ✜③❡♠♦s ♣❛r❛ ♠❛tr✐③❡s✿

f(T) =anTn+_{· · ·}+a1T +a0I.

❚❛♠❜é♠ ❞✐r❡♠♦s q✉❡T é ✉♠ ③❡r♦ ♦✉ r❛✐③ ❞❡f s❡f(T) = 0✳

❙❡ A é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡ T✱ ❡♥tã♦ f(A) é ❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡

f(T)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱f(T) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱f(A) = 0✳

❙❡❥❛ T :V _−→V ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ♥✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V s♦❜r❡ ✉♠ ❝♦r♣♦ K✳ ❯♠

❡s❝❛❧❛rλ_∈K é ❝❤❛♠❛❞♦ ❛✉t♦✈❛❧♦r ❞❡ _T✱ s❡ ❡①✐st❡ ✉♠ ✈❡t♦r ♥ã♦✲♥✉❧♦_{v ∈V}✱ ♣❛r❛ ♦ q✉❛❧

T(v) =λv.

❚♦❞♦ ✈❡t♦r ♥ã♦ ♥✉❧♦ q✉❡ s❛t✐s❢❛ç❛ ❡st❛ r❡❧❛çã♦ é ❝❤❛♠❛❞♦ ✉♠ ❛✉t♦✈❡t♦r ❞❡ T ❛ss♦❝✐❛❞♦

❛♦ ❛✉t♦✈❛❧♦rλ✳

◆♦t❡ q✉❡ ❝❛❞❛ ♠ú❧t✐♣❧♦ ❡s❝❛❧❛r kv ❞❡ ✉♠ ❛✉t♦✈❡t♦r v é t❛♠❜é♠ ✉♠ ❛✉t♦✈❡t♦r ❞❡ T✱

♣♦✐s

T(kv) = kT(v) = k(λv) =λ(kv).

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❛✉t♦✈❡t♦r❡s ❛ss♦❝✐❛❞♦s ❛λ é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ V✱ ❝❤❛♠❛❞♦ ❛✉t♦✲

❡s♣❛ç♦ ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦rλ✳

❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ❞á ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ✐♠♣♦rt❛♥t❡ ❞❡ ❛✉t♦✈❛❧♦r❡s✱ q✉❡ é ❢r❡q✉❡♥✲ t❡♠❡♥t❡ ✉s❛❞❛ ❝♦♠♦ s✉❛ ❞❡✜♥✐çã♦✳

❚❡♦r❡♠❛ ✶✳✷✳ ❙❡❥❛ T :V _−→V ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✳ ❊♥tã♦ λ_∈K é ✉♠ ❛✉t♦✈❛❧♦r ❞❡ _T

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♦♣❡r❛❞♦rλI₋T é s✐♥❣✉❧❛r✳ ❖ ❛✉t♦❡s♣❛ç♦ ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦r λ é

♦ ♥ú❝❧❡♦ ❞❡λI ₋T✳

Pr♦✈❛✳ λ é ✉♠ ❛✉t♦✈❛❧♦r ❞❡ T s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ✈❡t♦r ♥ã♦ ♥✉❧♦ v t❛❧ q✉❡

T(v) = λv_⇔(λI₋T)(v) = 0,

✐st♦ é✱ λI ₋T é s✐♥❣✉❧❛r✳ ❆❧é♠ ❞✐ss♦✱ v ❡stá ♥♦ ❛✉t♦❡s♣❛ç♦ ❞❡ λ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s

r❡❧❛çõ❡s ❛❝✐♠❛ ❛❝♦♥t❡❝❡♠✳ P♦rt❛♥t♦✱ v ❡stá ♥♦ ♥ú❝❧❡♦ ❞❡ λI ₋T✳

❙❡B é ✉♠❛ ❜❛s❡ ♦r❞❡♥❛❞❛ ❛r❜✐trár✐❛ ❞❡V ❡A= [T]B ✱ ❡♥tã♦ (λI−T)é ✐♥✈❡rsí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❛tr✐③ (λI₋A)é ✐♥✈❡rsí✈❡❧✳

❙❡❥❛ A é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n s♦❜r❡ ♦ ❝♦r♣♦ K✱ ✉♠ ❛✉t♦✈❛❧♦r ❞❡ ❆ ❡♠ K é ✉♠ ❡s❝❛❧❛r_λ ❡♠ K t❛❧ q✉❡ ❛ ♠❛tr✐③ _(λI−A) s❡❥❛ s✐♥❣✉❧❛r✱ ♦✉ s❡❥❛✱ ♥ã♦✲✐♥✈❡rtí✈❡❧✳

❆ ♠❛tr✐③

xIn₋A

é ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ A✳ ❙❡✉ ❞❡t❡r♠✐♥❛♥t❡✱

❞❡t(xI₋A),

q✉❡ é ✉♠ ♣♦❧✐♥ô♠✐♦ ❡♠x✱ pA(x)✱ ❞❡♥♦♠✐♥❛✲s❡ ♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ A✳

❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♣❛r❛ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡T s❡rã♦ ❛s r❛í③❡s ❞♦

♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ T✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✐st♦ ♥♦s ♠♦str❛ q✉❡ T ♥ã♦ ♣♦❞❡ t❡r ♠❛✐s

q✉❡n❛✉t♦✈❛❧♦r❡s ❞✐st✐♥t♦s✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡T ♣♦❞❡ ♥ã♦ t❡r ♥❡♥❤✉♠ ❛✉t♦✈❛❧♦r✳

❚❡♦r❡♠❛ ✶✳✸✳ ❙❡❥❛ A ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n s♦❜r❡ ✉♠ ❝♦r♣♦ K✳ ❯♠ ❡s❝❛❧❛r

λ _∈ K é ✉♠ ❛✉t♦✈❛❧♦r ❞❡ _A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ _λ é ✉♠❛ r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦

p(x) =❞❡t(xIn₋A) ❞❡ A✳

Pr♦✈❛✳ P❡❧♦ ❚❡♦r❡♠❛ ✭✶✳✸✮✱ λ é ✉♠ ❛✉t♦✈❛❧♦r ❞❡ A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ λI₋A é s✐♥❣✉❧❛r✱

♦✉ s❡❥❛✱ ❞❡t(λI ₋A) = 0✱ ✐st♦ é✱λ é ✉♠❛ r❛✐③ ❞❡ p✳

❊①❡♠♣❧♦ ✶✳✷✳ ❙❡❥❛ T ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r s♦❜r❡ R2 _{q✉❡ é r❡♣r❡s❡♥t❛❞♦ ❡♠ r❡❧❛çã♦ à ❜❛s❡}

♦r❞❡♥❛❞❛ ❝❛♥ô♥✐❝❛ ♣❡❧❛ ♠❛tr✐③

A=

"

0 ₋1 1 0

#

.

❊♥tã♦ ♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ T (♦✉ ❞❡ A) é

pA(x) = ❞❡t(xI₋A) =

x 1

−1 x

=x2 + 1.

✶✳✸✳ ❆❯❚❖❱❆▲❖❘❊❙ ❊ ❆❯❚❖❱❊❚❖❘❊❙

◆♦t❡ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ ❞♦ ❊①❡♠♣❧♦ ♥ã♦ ♣♦ss✉✐ r❛í③❡s r❡❛✐s✳ P♦rt❛♥t♦✱ T ♥ã♦ ♣♦ss✉❡♠

❛✉t♦✈❛❧♦r❡s✳ ❙❡U é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r s♦❜r❡ C2 q✉❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r _A ❡♠ r❡❧❛çã♦ à

❜❛s❡ ♦r❞❡♥❛❞❛ ❝❛♥ô♥✐❝❛✱ ❡♥tã♦ U ♣♦ss✉✐ ❞♦✐s ❛✉t♦✈❛❧♦r❡s✱ i ❡ ₋i✳ ❱❡♠♦s ❛q✉✐ ✉♠ ♣♦♥t♦

s✉t✐❧✳ ❆♦ ❞✐s❝✉t✐r♠♦s ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ ✉♠❛ ♠❛tr✐③ A✱ ♣r❡❝✐s❛♠♦s t♦♠❛r ♦ ❝✉✐❞❛❞♦ ❞❡

❡st✐♣✉❧❛r ♦ ❝♦r♣♦ ❡♥✈♦❧✈✐❞♦✳ ❆ ♠❛tr✐③ A ♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠ ✈❛❧♦r ❝❛r❛❝t❡ríst✐❝♦ ❡♠ R✱

♠❛s ♣♦ss✉✐ ❞♦✐s ✈❛❧♦r❡s ❝❛r❛❝t❡ríst✐❝♦s ❡♠C✳

❊①❡♠♣❧♦ ✶✳✸✳ ❙❡❥❛A ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ 3✱

A=



 

3 1 ₋1 2 2 ₋1 2 2 0



 

❖ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡A é

pA(x) = ❞❡t(xI3−A) =

x₋3 ₋1 1

−2 x₋2 1

−2 ₋2 x

=x3₋5x2 + 8x₋4 = (x₋1)(x₋2)2.

P♦rt❛♥t♦✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡A sã♦ ✶ ❡ ✷✳

▲❡♠❛ ✶✳✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ T(v) = λv ❡ q✉❡ f é q✉❛❧q✉❡r ♣♦❧✐♥ô♠✐♦ ❡♠ K_[x]✳ ❊♥tã♦

f(T)(v) = f(λ)v✱ ♦✉ s❡❥❛✱ f(λ) é ♦ ❛✉t♦✈❛❧♦r ❞❡ f(T) ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❡t♦r v✳

Pr♦✈❛✳ ❈♦♠♦ T(v) =λv t❡♠♦s q✉❡

T2(v) =T(T(v)) = T(λv) = λT(v) =λ2v.

❆ss✐♠✱ ✐♥❞✉t✐✈❛♠❡♥t❡✱ ♦❜t❡♠♦s

Tm_{(v) = λ}m_v,

∀ m_∈N_.

❆❣♦r❛✱ s❡❥❛

f(x) =a0+a1x+· · ·+anxn

✉♠ ♣♦❧✐♥ô♠✐♦ q✉❛❧q✉❡r✳ ❊♥tã♦

f(T)(v) =f(λ)v.

P♦rt❛♥t♦✱ f(λ) é ♦ ❛✉t♦✈❛❧♦r ❞❡f(T) ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❡t♦r v✳

❈❛♣ít✉❧♦ ✷

▼❛tr✐③ ❈✐r❝✉❧❛♥t❡

▼❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s sã♦ ♣r❡❞♦♠✐♥❛♥t❡s ❡♠ ♠✉✐t♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛✳ ❊ss❛s ♠❛tr✐✲ ③❡s ❛♣❛r❡❝❡♠ ♥❛t✉r❛❧♠❡♥t❡ ❡♠ ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✱ ♦♥❞❡ ❛s r❛í③❡s ❞❛ ✉♥✐❞❛❞❡ ❞❡s❡♠♣❡✲ ♥❤❛♠ ✉♠ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧✱ ❡ ❛q✉✐ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♠❛s ❞❛s r❛③õ❡s ❞❡ss❛ ✐♠♣♦rtâ♥❝✐❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡❧❛s sã♦ ♦♥✐♣r❡s❡♥t❡s✱ ♠✉✐t♦s ❢❛t♦s s♦❜r❡ ❡ss❛s ♠❛tr✐③❡s ♣♦❞❡♠ s❡r ♣r♦✈❛❞❛s ✉s❛♥❞♦ ❛♣❡♥❛s á❧❣❡❜r❛ ❧✐♥❡❛r ❜ás✐❝❛✳ ■ss♦ t♦r♥❛ ❛ ár❡❛ ❜❛st❛♥t❡ ❛❝❡ssí✈❡❧ ♣❛r❛ ❛❧✉♥♦s ❞❡ ❣r❛❞✉❛çã♦ à ♣r♦❝✉r❛ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ♣❡sq✉✐s❛ ❡✴♦✉ ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❡♠ ❜✉s❝❛ ❞❡ t❡♠❛s ❝♦♠ ✐♥t❡r❡ss❡ ❡①❝❧✉s✐✈♦ ♣❛r❛ ❛♣r❡s❡♥t❛r ❛♦s s❡✉s ❛❧✉♥♦s✳

✷✳✶ ❈♦♥❝❡✐t♦s ❡ Pr♦♣r✐❡❞❛❞❡s

❊♠ t✉❞♦ q✉❡ s❡❣✉❡ n _∈ N✱ ❝♦♠ _{n ≥ 2}✳ ◆♦ss♦s ♦❜❥❡t♦s ❞❡ ❡st✉❞♦ ♥❡st❡ ❝❛♣ít✉❧♦ sã♦✿

♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ Cn _{❡ ♦ ❛♥❡❧ ❞❛s ♠❛tr✐③❡s ❝♦♠♣❧❡①❛s Mn(C)✳ ❊st✉❞❛r❡♠♦s ❛}

♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s M _∈ Mn(C₎ ♣♦r ✈❡t♦r❡s _{v ∈} Cn_{✳ ❆ ❡st❡ r❡s♣❡✐t♦✱ ♣♦❞❡♠♦s}

✐❞❡♥t✐✜❝❛r ♦ ✈❡t♦rv ❝♦♠♦ ✉♠ ✈❡t♦r ❝♦❧✉♥❛ ✭♠❛tr✐③ ❝♦❧✉♥❛✮✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦r ✈❡③❡s✱ é út✐❧

♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❡ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ t✐♣♦❣r❛✜❝❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❛r

v = (v0, v1, v2, . . . , vn−1)∈Cn ←→V =vT ∈Mn×1(C).

❝♦♠♦ ✉♠ ✈❡t♦r ❝♦❧✉♥❛✳ ❙❡❥❛

E =_{e0, e1, . . . , en−1}

❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❝❛♥ô♥✐❝❛ ♣❛r❛ Cn_{✱ ❡♠ q✉❡}

ei = (δi,0, . . . , δi,n−1), i = 0, . . . n−1,

❡ δi,j é ♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✶

δi,j =

(

1, s❡ i=j

0, s❡ i₆=j.

❊♥tã♦ ❝❛❞❛ ✈❡t♦r v ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ♠♦❞♦ ú♥✐❝♦ s♦❜ ❛ ❢♦r♠❛

v = (v0, v1, v2, . . . , vn−1) =

n−1 X

i=0

viei,

s❡♥❞♦ ♦s ❡s❝❛❧❛r❡s vi ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ v ❝♦♠ r❡s♣❡✐t♦ à ❜❛s❡_E✳

❙❡❥❛♠ S=_{0,1, . . . , n₋1_} ❡σ :S _→S ❛ ♣❡r♠✉t❛çã♦ ❞❡✜♥✐❞❛ ♣♦r σ(i) =

(

i₋1, s❡ 1_≤i_≤n₋1

n₋1, s❡ i= 0.

❱❛♠♦s ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦r ❞❡s❧♦❝❛♠❡♥t♦ ❝í❝❧✐❝♦ Tσ :Cn_−→Cn _{❛ss♦❝✐❛❞♦ ❛ σ ♣♦r} Tσ(v0, v1, v2, . . . , vn−1) = (vσ(0), vσ(1), vσ(2), . . . , vσ(n−1)).

◆♦t❡ q✉❡

T_σ2(v0, v1, v2, . . . , vn−1) = (vσ2

(0), vσ2

(1), vσ2

(2), . . . , vσ2

(n−1)).

❊♠ ❣❡r❛❧✱

Tk

σ =Tσk ❡ Tn

σ =Tσn =I.

❙❡v _∈Cn_{✱ ❝♦♠ v 6= 0✱ ❡♥tã♦ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡}

B =_{v, T(v), . . . , Tn−1_(v)

}

é ✉♠❛ ❜❛s❡ ♦r❞❡♥❛❞❛ ♣❛r❛ Cn_✳

✶_{▲❡♦♣♦❧❞ ❑r♦♥❡❝❦❡r ❡st✉❞♦✉ ❡♠ ❇❡r❧✐♠ ❡ ♦❜t❡✈❡ ♦ ❣r❛✉ ❞❡ ❞♦✉t♦r ❡♠ ✶✽✹✺ ❝♦♠ ✉♠❛ t❡s❡ s♦❜r❡}

❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❆s s✉❛s ♣r✐♥❝✐♣❛✐s ❝♦♥tr✐❜✉✐çõ❡s ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ❢♦r❛♠ ♥♦ ❝❛♠♣♦ ❞❛ á❧❣❡❜r❛ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s✳

✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❊ P❘❖P❘■❊❉❆❉❊❙

❙❡❥❛♠

q(x) = 1 + 2x+x2+ 3x3 _∈R_[x] ❡ _{W =}

     

0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0

      ,

❡♠ q✉❡ W é ❛ tr❛♥s♣♦st❛ ❞❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡ Tσ ❡♠ r❡❧❛çã♦ ❛ ❜❛s❡ ♦r❞❡♥❛❞❛

❝❛♥ô♥✐❝❛ ♣❛r❛ R4✱ ♣♦✐s

Tσ(e0) = e1, Tσ(e1) = e2, Tσ(e2) =e3 ❡ Tσ(e3) = e0.

❊♥tã♦ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡

C =q(W) =

     

1 2 1 3 3 1 2 1 1 3 1 2 2 1 3 1

     

é ✉♠❛ ♠❛tr✐③ ♦❜t✐❞❛ ❞❡s❧♦❝❛♥❞♦ ❝✐❝❧✐❝❛♠❡♥t❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ✉♠❛ ✉♥✐❞❛❞❡ ♣❛r❛ à ❞✐r❡✐t❛✳ ■st♦ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿

❯♠❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡ C =❝✐r❝_{v_}❛ss♦❝✐❛❞❛ ❛ ✉♠ ✈❡t♦rv _∈Cné ❛ ♠❛tr✐③_n×n❝✉❥❛s

❧✐♥❤❛s sã♦ ❞❛❞❛s ♣♦r ✐t❡r❛çõ❡s ❞♦ ♦♣❡r❛❞♦r ❞❡s❧♦❝❛♠❡♥t♦ ❝í❝❧✐❝♦ ❛❣✐♥❞♦ ❡♠ s✉❛ k✲és✐♠❛

❧✐♥❤❛✱Tk−1

σ (v), k= 1, . . . , n✿

C =         

v0 v1 v3 · · · vn−1

vn−1 v0 v1 · · · vn−2

vn−2 vn−1 v0 · · · vn−3

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

v1 v2 v3 · · · v0          .

❆ ♦r❞❡♥❛çã♦ ❡s❝♦❧❤✐❞❛

C=❝✐r❝(v0, v1, v2, . . . , vn−1)

é ❛♣❡♥❛s ✉♠❛ ❝♦♥✈❡♥çã♦ ❛❞❡q✉❛❞❛ ♣❛r❛ ✐♠♣♦r ✉♠❛ ♥♦t❛çã♦ ❝í❝❧✐❝❛ ❡♠ q✉❡ ♦s ♥ú♠❡r♦svk

❞❡✈❡♠ r❡❛❧♠❡♥t❡ ♦❜❡❞❡❝❡r✳ ❖❜s❡r✈❡ q✉❡C é ❛ ♠❛tr✐③ ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡ ❞❛ ❜❛s❡ _B♣❛r❛ ❛

❜❛s❡ E✳

❊①❡♠♣❧♦ ✷✳✶✳ ❆ ♣❛rt✐r ❞♦ ✈❡t♦r v = (a, b, c)_∈R3✱ ♣♦❞❡♠♦s ❣❡r❛r ❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡

C3 = 

 

a b c c a b b c a



 .

❱❡r❡♠♦s q✉❡ ❛s ♠❛tr✐③❡s ❝✐r❝✉❧❛♥t❡s ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡♠ ✈ár✐❛s ❛♣❧✐❝❛çõ❡s✱ ❞❡♥tr❡ ❛s q✉❛✐s ❛ q✉❡ ♠❛✐s ♥♦s ❝❤❛♠❛ ❛ ❛t❡♥çã♦ é ❛ ❢♦r♠❛ s✐♠♣❧❡s ❝♦♠♦ s❡ ❝❛❧❝✉❧❛ s❡✉s ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s ✉t✐❧✐③❛♥❞♦ r❛í③❡sn✲és✐♠❛s ❞❛ ✉♥✐❞❛❞❡✳

✷✳✷ ▼❛tr✐③❡s ❞❡ P❡r♠✉t❛çõ❡s

❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡ é ❛ ♠❛tr✐③ ❞❡ ♣❡r♠✉t❛çã♦

W =❝✐r❝_{ei_},

❛ q✉❛❧ é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ s✉❛ ❧✐♥❤❛ s✉♣❡r✐♦r ♠✉❞❛❞❛ ♣❛r❛ ❛ ♣❛rt❡ ✐♥❢❡r✐♦r✳

❊①❡♠♣❧♦ ✷✳✷✳ ◗✉❡♠ é W =❝✐r❝_{e1}✱ q✉❛♥❞♦ n = 3? ❈♦♠♦ e1 = (0,1,0) t❡♠♦s q✉❡

W =



 

0 1 0 0 0 1 1 0 0



 .

◆♦t❡ q✉❡ s❡ Wi =❝✐r❝_{ei_} ❡Wj =❝✐r❝_{ej_}✱ ❡♥tã♦ Wi_·Wj =Wi+j ❡ Wi =Wi

❡♠ q✉❡ ♦s í♥❞✐❝❡s sã♦ ❧✐❞♦s ♠ó❞✉❧♦n✳ ➱ ❝❧❛r♦ q✉❡ W0 =I ❡

Wt=W−1,

♣♦✐s

W Wt₌



 

0 1 0 0 0 1 1 0 0

     

0 0 1 1 0 0 0 1 0

  =   

1 0 0 0 1 0 0 0 1





=I.

✷✳✷✳ ▼❆❚❘■❩❊❙ ❉❊ P❊❘▼❯❚❆➬Õ❊❙

◆❡st❡ ❝❛s♦✱W é ♠❛tr✐③ ♦rt♦❣♦♥❛❧✱ ♦✉ s❡❥❛✱ W WT _{=I✳ ❖❜s❡r✈❡ q✉❡} C =



 

a b c c a b b c a





=aI+bW +cW

2_.

❈♦♠♦W3 _{=I t❡♠♦s q✉❡W é ✉♠❛ r❛✐③ ❞❛ ❡q✉❛çã♦ x}3_{−1 = 0✳ ❆ss✐♠✱ ❛ ❡q✉❛çã♦}

xn

−1 = 0

❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✳ ❆s r❛í③❡s ❞❡ss❛ ❡q✉❛çã♦ sã♦ ❞❡♥♦♠✐✲ ♥❛❞❛s r❛í③❡s n✲és✐♠❛s ❞❛ ✉♥✐❞❛❞❡✳ ➱ ❝♦♠✉♠ ❞❡✜♥✐r ❡ss❛s r❛í③❡s ♣♦r

ω= cos

2π n

+isin

2π n

=e2πin

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r✷_{✳ ◆❡st❡ ❝❛s♦✱ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦x}n

−1 = 0 sã♦ ❞❛❞❛s

♣❡❧❛s ♣♦tê♥❝✐❛s n✲és✐♠❛s ❞❡ ω ❡✱ ♣❛r❛ ❝❛❞❛k _∈N✱ ❝♦♠_{k = 0,1, . . . , n−1}✱ t❡♠✲s❡

ωk =e2πikn .

P♦rt❛♥t♦✱

1, ω, ω2, . . . , ωn−1

sã♦ ❛s r❛í③❡sn✲és✐♠❛s ✭❞✐st✐♥t❛s✮ ❞❡xn

−1 = 0✳

P❛r❛ ❡①♣❧♦r❛r ✉♠❛ ♦✉tr❛ ❜❛s❡ ♣❛r❛ Cn✱ ❝♦♥s✐❞❡r❡♠♦s ♦s ✈❡t♦r❡s ✭❛✉t♦✈❡t♦r❡s ♥♦r♠❛✲

❧✐③❛❞♦s ❞❡ W✮

xk = √1

n(1, ω k

, ω2k, . . . , ω(n−1)k)_∈Cn_{, k= 0,1, . . . , n−1,}

❞❡ ♠♦❞♦ q✉❡ ♦s ✈❡t♦r❡s xk ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ Cn_{✳ ❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ k✱ ♣♦❞❡♠♦s}

❝♦♥str✉✐r ✉♠❛ ♠❛tr✐③ ❡s♣❡❝✐❛❧ ❛ ♣❛rt✐r ❞❛ ▼❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡✱ ❝♦❧♦❝❛♥❞♦ ♦s ✈❡t♦r❡s

✷_{❆ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r✱ ❝✉❥♦ ♥♦♠❡ é ✉♠❛ ❤♦♠❡♥❛❣❡♠ ❛ ▲❡♦♥❤❛r❞ ❊✉❧❡r✱ é ✉♠❛ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ ❞❛}

ár❡❛ ❡s♣❡❝í✜❝❛ ❞❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ q✉❡ ♠♦str❛ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳

xk ❝♦♠♦ ❝♦❧✉♥❛s

Q= √1

n         

1 1 1 _{· · ·} 1 1 ω ω2 _{· · · ω}n−1

1 ω2 _ω4 _{· · · ω}2(n−1)

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

1 ωn−1 _ω2(n−1) _{· · · ω}(n−1)2

         .

❉❛❞❛ A _∈ Mn(C₎✱ ❛ tr❛♥s♣♦st❛ ❝♦♥❥✉❣❛❞❛ ❞❡ _A é ❞❡♥♦♠✐♥❛❞❛ ❍❡r♠✐t✐❛♥❛ ❡ ❛q✉✐

❞❡♥♦t❛❞❛ ♣♦r A∗_✳

❯♠❛ ♠❛tr✐③ A _∈Mn(C₎é✿

• ❍❡r♠✐t✐❛♥❛✱ s❡ A=A∗_✳

• ❯♥✐tár✐❛✱ s❡ A−1 _=A∗_✳

• ◆♦r♠❛❧✱ s❡AA∗ _=A∗_A✳

❈❧❛r❛♠❡♥t❡✱ ♠❛tr✐③❡s ✉♥✐tár✐❛s ❡ ❍❡r♠✐t✐❛♥❛s sã♦ s❡♠♣r❡ ♥♦r♠❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♠♦ W é ✉♠❛ ♠❛tr✐③ r❡❛❧ ❡ W∗ _=WT _{t❡♠♦s q✉❡ W é ✉♥✐tár✐❛ ❡✱ ♣♦rt❛♥t♦✱ ♥♦r♠❛❧✳}

❯♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ q✉❡ ❝❛r❛❝t❡r✐③❛ ♠❛tr✐③❡s ♥♦r♠❛✐s é q✉❡ ❡❧❛s sã♦ ✉♥✐t❛r✐❛✲ ♠❡♥t❡ ❞✐❛❣♦♥❛❧✐③á✈❡✐s✱ ♦✉ s❡❥❛✱ A é ♥♦r♠❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r ✉♠❛ ♠❛tr✐③ ✉♥✐tár✐❛ U t❛❧ q✉❡ U∗_{AU s❡❥❛ ❞✐❛❣♦♥❛❧✳}

P❛ss❛♠♦s ❛ ❝♦♥str✉✐r ✉♠❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ♣❛r❛ W✳ ❈♦♠♦ W é ✉♠❛ ♠❛tr✐③ ❞❡ ♣❡r♠✉✲

t❛çã♦✱ ♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ W é✿

P(x) = ❞❡t

        

x ₋1 0 _{· · ·} 0 0 x ₋1 _{· · ·} 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 0 _{· · · −}1

−1 0 0 _{· · ·} x

        

= (₋1)n+1_(−1)(−1)n−1_{+ (−1)}n+n_x

·xn−1

= xn

−1.

.

P♦rt❛♥t♦✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ W sã♦ ❛s r❛í③❡s n✲és✐♠❛s ❞❛ ✉♥✐❞❛❞❡✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡

v(ω) = (1, ω, ω2, . . . , ωn−1)

✷✳✷✳ ▼❆❚❘■❩❊❙ ❉❊ P❊❘▼❯❚❆➬Õ❊❙

é ♦ ❛✉t♦✈❡t♦r ❞❡ W ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦r ω✳ ▼❛✐s ❣❡r❛❧♠❡♥t❡✱ ❝♦♠♦

ωk _=e2kπin , k = 0,1,2, . . . , n−1

t❡♠♦s q✉❡ ♦s ❛✉t♦✈❡t♦r❡s ❞❡W sã♦✿

v(ω0_{), v(ω}1_{), . . . , v(ω}n−1₎

❛ss♦❝✐❛❞♦s ❛♦s ❛✉t♦✈❛❧♦r❡s

ω0 _{= 1, ω}1_{, . . . , ω}n−1_.

❙❡❥❛♠ Q ❛ ♠❛tr✐③ ❝✉❥❛s ❛s ❝♦❧✉♥❛s sã♦ ❡ss❡s ❛✉t♦✈❡t♦r❡s ❡ D ❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❝✉❥♦s

❡❧❡♠❡♥t♦s ❞✐❛❣♦♥❛✐s s❡❥❛♠ ♦s ❛✉t♦✈❛❧♦r❡s ❛ss♦❝✐❛❞♦s✳ ❊♥tã♦

W Q=QD. ✭✷✳✶✮

➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡

Q−1 = Q

∗

n

❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱

1 √ n Q

é ✉♠❛ ♠❛tr✐③ ✉♥✐tár✐❛✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ ♣♦❞❡ s❡r ❡s❝r✐t♦ s♦❜ ❛ ❢♦r♠❛

W = 1 √ nQ D 1 √ nQ ∗ ✭✷✳✷✮

q✉❡ é ✉♠❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ✉♥✐tár✐❛ ❞❡ W✳

❊①❡♠♣❧♦ ✷✳✸✳ ❙❡❥❛

Q=



 

1 1 1 1 ω ω2

1 ω2 _ω 

 ,

❝♦♠ ω ❛ r❛✐③ ❝ú❜✐❝❛ ❞❛ ✉♥✐❞❛❞❡✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ W é ❞✐❛❣♦♥❛❧✐③á✈❡❧✱ ♦✉ s❡❥❛✱

❙♦❧✉çã♦✳ ❈♦♠♦

ω =₋1 2+

√

3 2 i

t❡♠♦s q✉❡ ❞❡t(Q) =₋3i√3₆= 0✳ ❆ss✐♠✱ Q−1 _{❡①✐st❡✳ ◆♦t❡ q✉❡}

Q∗Q =



 

1 1 1 1 ω2 _ω

1 ω ω2      

1 1 1 1 ω ω2

1 ω2 _ω    =   

3 1 +ω+ω2 _{1 +ω+ω}2

1 +ω+ω2 _{3 1 +ω+ω}2

1 +ω+ω2 _{1 +ω+ω}2 ₃

   =   

3 0 0 0 3 0 0 0 3



 ,

♣♦✐s 1 +ω+ω2 _{= 0✱ ❞❡ ♠♦❞♦ q✉❡}

Q∗Q= 3I ❡ Q−1 = 1 3Q ∗_. P♦rt❛♥t♦✱ W = Q √ 3 D Q √ 3 ∗ ,

q✉❡ é ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

❱❛♠♦s ❡st❡♥❞❡r ❡st❡s r❡s✉❧t❛❞♦s✱ ✈✐❛ ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s✱ ♣❛r❛ ♠❛tr✐③❡s ❝✐r❝✉✲ ❧❛♥t❡s ❣❡r❛✐s✳ ❈♦♠♦ ✈✐♠♦s ✉♠❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦

C =❝✐r❝_{v_}=

n−1 X

i=0

viWi.

❖ ♣♦❧✐♥ô♠✐♦

q(x) =qC(x) =

n−1 X

i=0

vixi

∈R_[x] ✭✷✳✸✮

❛ss♦❝✐❛❞♦ ❛ ♠❛tr✐③ ❝✐r❝✉❧❛♥t❡ C =❝✐r❝_{v_}é ❞❡♥♦♠✐♥❛❞♦ ♦ r❡♣r❡s❡♥t❛♥t❡ ❞❡ C=❝✐r❝_{v_}✳