Um estudo introdutório da Teoria de Grafos através de matrizes

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❯♠ ❊st✉❞♦ ■♥tr♦❞✉tór✐♦ ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s

❆tr❛✈és ❞❡ ▼❛tr✐③❡s

❉✐❡❣♦ ❘♦❞r✐❣✉❡s ●♦♥ç❛❧✈❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❚❤✐❛❣♦ ❞❡ ▼❡❧♦

✺✶✶✳✺ ●✻✸✺❡

●♦♥ç❛❧✈❡s✱ ❉✐❡❣♦ ❘♦❞r✐❣✉❡s

❯♠ ❊st✉❞♦ ■♥tr♦❞✉tór✐♦ ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s ❆tr❛✈és ❞❡ ▼❛tr✐③❡s✴ ❉✐❡❣♦ ❘♦❞r✐❣✉❡s ●♦♥ç❛❧✈❡s✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✹✳

✺✵ ❢✳✿ ✜❣✳✱ t❛❜✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r✿ ❚❤✐❛❣♦ ❞❡ ▼❡❧♦

✶✳ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s✳ ✷✳ ●r❛❢♦✳ ✸✳ ▼❛tr✐③✳ ✹✳ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❉✐❡❣♦ ❘♦❞r✐❣✉❡s ●♦♥ç❛❧✈❡s

❯♠ ❊st✉❞♦ ■♥tr♦❞✉tór✐♦ ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s ❆tr❛✈és ❞❡

▼❛tr✐③❡s

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❚❤✐❛❣♦ ❞❡ ▼❡❧♦ ❖r✐❡♥t❛❞♦r

Pr♦❢❛✳ ❉r❛✳ ❊❧ír✐s ❈r✐st✐♥❛ ❘✐③③✐♦❧❧✐ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯◆❊❙P

Pr♦❢✳ ❉r✳ ❚♦♠❛s ❊❞s♦♥ ❞❡ ❇❛rr♦s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❙❈❆❘

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♠ã❡✱ ❉❡❧❤❛❡✉♥✐❝❡✱ ❡ ❛♦ ♠❡✉ ♣❛✐✱ ❏♦sé ❘❛✐♠✉♥❞♦✱ q✉❡ s❡♠♣r❡ ❜✉s❝❛r❛♠✱ ❝♦♠ ♠✉✐t♦ ❡s❢♦rç♦✱ ♣r♦♣✐❝✐❛r ♦ ♠❡❧❤♦r ♣❛r❛ ♦s ✜❧❤♦s✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❚❛❧✐t❛ q✉❡✱ ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ✶✸ ❛♥♦s✱ t❡♠ ♠❡ ❛♣♦✐❛❞♦ ❡♠ t♦❞❛s ❛s ❞❡❝✐sõ❡s✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ❘❛❢❛❡❧✱ ❊✇❡rt♦♥ ❡ ❚❤✐❛❣♦ q✉❡ ♠❡ ✐♥s♣✐r❛r❛♠ ❛ s❡♠♣r❡ ❢❛③❡r ♦ ♠❡❧❤♦r✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ s♦❣r❛ ❆❜✐❣❛✐❧ ✭✐♥ ♠❡♠♦r✐❛♠✮ ♣♦r t♦❞♦ ❛✉①✐❧✐♦ ♣r❡st❛❞♦✱ ❡s♣❡❝✐✲ ❛❧♠❡♥t❡ ♥♦s ♣r✐♠❡✐r♦s ❛♥♦s ❞❡ ❣r❛❞✉❛çã♦✳

❆❣r❛❞❡ç♦ à t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠✳ ❆❣r❛❞❡ç♦ à ❡q✉✐♣❡ ❣❡st♦r❛ ❞❡ ❞❛ ❊s❝♦❧❛ ▼✉♥✐❝✐♣❛❧ ■♥t❡❣r❛çã♦✱ ❞❡ ❱✐♥❤❡❞♦✱ ♣❡❧❛ ❡♥♦r♠❡ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♣♦✐♦✳

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❡ r❡❧❛❝✐♦♥á✲❧♦s ❝♦♠ ❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s✱ ♣♦r ♠❡✐♦ ❞❡ ❡①❡♠♣❧♦s✱ s❡♠♣r❡ q✉❡ ♣♦ssí✈❡❧✳ ❆ ❢❡rr❛♠❡♥t❛ ❜ás✐❝❛ ♣❛r❛ ✐ss♦ é ❛ t❡♦r✐❛ ❞❡ ♠❛tr✐③❡s✳

❆❜str❛❝t

❚❤❡ ❛✐♠ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t s♦♠❡ ❡❧❡♠❡♥t❛r② r❡s✉❧ts ❢r♦♠ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❛♥❞ t♦ r❡❧❛t❡ t❤❡♠ ✇✐t❤ ●r❛♣❤ ❚❤❡♦r②✱ ♠❛❦✐♥❣ ✉s❡ ♦❢ ❡①❛♠♣❧❡s ✐❢ ♣♦ss✐❜❧❡✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✽

✷ ▼❛tr✐③❡s ❡ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✾

✷✳✶ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ❈♦♥❝❡✐t♦s ❊❧❡♠❡♥t❛r❡s ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛❢♦s ✷✶

✸✳✶ ●r❛❢♦s ❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✷ ❈✐❝❧♦s ❢✉♥❞❛♠❡♥t❛✐s ❡ ❝♦rt❡s ❢✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹ ❙✉❣❡stã♦ ❞❡ ❆✉❧❛s ✹✹

✹✳✶ ❆t✐✈✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✶ ■♥tr♦❞✉çã♦

❆ ♣r♦♣♦st❛ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r❛✲ ❢♦s✱ ❜❡♠ ❝♦♠♦ r❡❧❛❝✐♦♥á✲❧♦s ❝♦♠ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳

❖ ❡st✉❞♦ ❝♦♠❡ç❛ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s à ♠❛tr✐③❡s✱ ♣❛rt✐♥❞♦ ❞❛s ♠❛✐s s✐♠♣❧❡s ❞❡✜♥✐çõ❡s✱ ❡♠ ❞✐r❡çã♦ ❛ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✱ ♥ã♦ ❡①✐❣✐♥❞♦ ❞♦ ❧❡✐t♦r ✈❛st❛ ❡①♣❡r✐ê♥❝✐❛ ♠❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❛❝♦♠♣❛♥❤❛♠❡♥t♦ ❞♦ ❛ss✉♥t♦ ❛♣r❡s❡♥t❛❞♦✳ ❆ ✐❞❡✐❛ ❞❡ s❡ tr❛❜❛❧❤❛r ❝♦♠ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝r✐❛r ✉♠❛ ❝♦♥❡①ã♦ ❡♥tr❡ ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞❛s✱ ✈á❧✐❞♦s ♣❛r❛ tr❛♥s❢♦r♠❛çõ❡s✱ ❡ ❡st❡♥❞ê✲❧♦s ♣❛r❛ ♠❛tr✐③❡s✱ t❛❧ ❝♦♠♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ s✉❜❡s♣❛ç♦ ✐♠❛❣❡♠ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❡ ♦ s✉❜❡s♣❛ç♦ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡ss❛ tr❛♥s❢♦r♠❛çã♦✳

❆♣ós ❡ss❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❞✐r❡❝✐♦♥❛♠♦s ♥♦ss♦ ❡st✉❞♦ ♥♦ s❡♥t✐❞♦ ❞❡ ❞❡✜♥✐r♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❣r❛❢♦s✱ ♦♣t❛♥❞♦ ❡♠ ❢❛③ê✲❧♦s ❞❡ ♠♦❞♦ s✉❝✐♥t♦ ❡ s✐♠♣❧✐✜❝❛❞♦✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ss❡ tr❛❜❛❧❤♦ ❢♦✐ ♦ ❝✉✐❞❛❞♦ ❡♠ t❡♥t❛r ❡①❡♠♣❧✐✜❝❛r r❡s✉❧t❛❞♦s q✉❡ ❢♦ss❡♠✱ ❡♠ ✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦✱ ❞✐❢í❝❡✐s✳ ❆❧é♠ ❞✐ss♦✱ ♣♦r s❡ tr❛t❛r ❞❡ ✉♠ t❡①t♦ ❜ás✐❝♦✱ ❡♥t❡♥❞❡♠♦s q✉❡ ♦ r❡❝✉rs♦ ❞♦s ❡①❡♠♣❧♦s ❝♦♥st✐t✉❡♠ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛ ❞✐❞át✐❝❛ ❞♦ t❡①t♦✳

❋✐♥❛❧♠❡♥t❡ t❡♠♦s ❛ ✏❝♦♥❡①ã♦✑ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❝♦♠ ❛ t❡♦r✐❛ ❞❡ ❣r❛❢♦s✱ ♦ q✉❡ é ♣♦ssí✈❡❧ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❡ ♠❛tr✐③ ❞❡ ✐♥❝✐❞ê♥❝✐❛ ❞❡ ✉♠ ❣r❛❢♦✳ ▼✉✐t♦s ❞♦s r❡s✉❧t❛❞♦s✱ ❡♠❜♦r❛ ❡❧❡♠❡♥t❛r❡s✱ ♣♦ss✉❡♠ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❛♣❧✐❝❛çã♦ ❞♦ ❡st✉❞♦ ❞❡ ❋❧✉①♦ ❡ ❘❡❞❡s✳ ❆ ❜❡❧❡③❛ ❞❡ss❡s r❡s✉❧t❛❞♦s s❡ ❡♥❝♦♥tr❛ ❡♠ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡ ♥❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✐♥t❡r♣r❡t❛çõ❡s q✉❡ s✉r❣❡♠ ❞❡❧❡s✳

❊ss❡ t❡①t♦ ♥ã♦ ❡①✐❣❡ ❞♦ ❧❡✐t♦r ✉♠❛ ✈❛st❛ ❡①♣❡r✐ê♥❝✐❛ ♠❛t❡♠át✐❝❛ s♦❜r❡ ♦s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s✱ ♣♦✐s tr❛t❛✲s❡ ❞❡ ✉♠❛ ✏♣r✐♠❡✐r❛ ❧❡✐t✉r❛✑ s♦❜r❡ ♦ t❡♠❛✳

✷ ▼❛tr✐③❡s ❡ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s

❆ ♣r♦♣♦st❛ ❞❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s r❡❧❛❝✐♦♥❛❞♦s ❛♦s ❝♦♥❝❡✐t♦s ❞❡ ♠❛tr✐③❡s ❡ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✳ ▼✉✐t♦s ❞♦s r❡s✉❧t❛❞♦s q✉❡ ❛q✉✐ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ q✉❛❧q✉❡r ❧✐✈r♦ ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❬✶❪✳

✷✳✶ ▼❛tr✐③❡s

❈❤❛♠❛♠♦s ♠❛tr✐③ ✉♠❛ t❛❜❡❧❛ ❞❡ ❡❧❡♠❡♥t♦s ❞✐s♣♦st♦s ❡♠ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✳ P♦❞❡♠♦s ❛tr✐❜✉✐r s✐❣♥✐✜❝❛❞♦ ❛s ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✳

❘❡♣r❡s❡♥t❛♠♦s ✉♠❛ ♠❛tr✐③ ❞❡m ❧✐♥❤❛s ❡n ❝♦❧✉♥❛s ♣♦r✿

Am×n=



    

a11 a12 · · · a1n

a21 a22 · · · a2n

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn



    

= [aij]_m×n

❉❡✜♥✐çã♦ ✷✳✶✳ ❉✉❛s ♠❛tr✐③❡s Am×n = [aij] ❡Br×s = [bij]sã♦ ✐❣✉❛✐s✱ A=B✱ s❡ ❡st❛s

tê♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s (m = r) ❡ ❝♦❧✉♥❛s (n =s) ❡ t♦❞♦s ♦s s❡✉s ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ✐❣✉❛✐s (aij =bij)✳

P♦r ❡①❡♠♣❧♦✱ ❛s ♠❛tr✐③❡s ❛❜❛✐①♦ sã♦ ✐❣✉❛✐s✱ ♣♦ré♠ ❝♦♠ r❡♣r❡s❡♥t❛çõ❡s ❞✐st✐♥t❛s✳

"

5−2 _{0 cos 45}➦

1 4 e

iπ ₄₈

#

= "

1 25 0

√

2 2

2−2 _{−1 48}

#

❚✐♣♦s ❞❡ ▼❛tr✐③❡s

❆❧❣✉♠❛s ♠❛tr✐③❡s ❛♣r❡s❡♥t❛♠ t✐♣♦s ❡s♣❡❝✐❛✐s ❞❡ ❡str✉t✉r❛s ❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛✐s✳ ➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ❡ss❛s ♠❛tr✐③❡s ❛♣❛r❡❝❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡ ❣r❛❢♦s ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ s❡ ❢❛③ ♥❡❝❡ssár✐❛✳

◆♦s ❝❛s♦s ❛ s❡❣✉✐r ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③An×m ❝♦♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✳

▼❛tr✐③ ◗✉❛❞r❛❞❛✿ é ❛q✉❡❧❛ ❡♠ q✉❡ m =n✱ ♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ❝♦✐♥❝✐❞❡✳ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ An×n t❛♠❜é♠ é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞❡ ♦r❞❡♠ n✳

▼❛tr✐③❡s ✶✵

▼❛tr✐③ ◆✉❧❛✿ é ❛q✉❡❧❛ ❡♠ q✉❡ aij = 0✱ ♣❛r❛ t♦❞♦ i ❡ j✳ ❆♦ ❧♦♥❣♦ ❞❡ ♥♦ss♦ t❡①t♦

r❡♣r❡s❡♥t❛r❡♠♦s ❡ss❛ ♠❛tr✐③ ♣♦r 0n×m✱ ♣❛r❛ ❞❡✐①❛r ❝❧❛r♦ ❛ ♦r❞❡♠ ❞❛ ♠❛tr✐③ ❡✱ q✉❛♥❞♦

♥ã♦ ❤♦✉✈❡r ❛♠❜✐❣✉✐❞❛❞❡✱ s✐♠♣❧❡s♠❡♥t❡ ♣♦r 0✳

▼❛tr✐③ ❈♦❧✉♥❛✿ é ❛q✉❡❧❛ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛✱ ♦✉ s❡❥❛ m= 1.❚❛♠❜é♠ é ❝♦♠✉♠ ♥♦s r❡❢❡r✐r♠♦s ❛ ✉♠❛ ♠❛tr✐③ ❝♦❧✉♥❛ ❝♦♠♦ ✉♠ ✈❡t♦r ❝♦❧✉♥❛✳

▼❛tr✐③ ▲✐♥❤❛✿ é ❛q✉❡❧❛ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ ❧✐♥❤❛✱ ♦✉ s❡❥❛✱ n = 1. ❚❛♠❜é♠ é ❝♦♠✉♠ ♥♦s r❡❢❡r✐r♠♦s ❛ ✉♠❛ ♠❛tr✐③ ❧✐♥❤❛ ❝♦♠♦ ✉♠ ✈❡t♦r ❧✐♥❤❛✳

▼❛tr✐③ ❉✐❛❣♦♥❛❧✿ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♦♥❞❡ aij = 0 ♣❛r❛ i6=j✳ ❆ ♠❛✐s ✐♠♣♦r✲

t❛♥t❡ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♦♥❞❡ aij = 0 ♣❛r❛ i 6= j ❡ aij = 1 ♣❛r❛

i=j✳ ❆ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n s❡rá r❡♣r❡s❡♥t❛❞❛ ♣♦r In✳

▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ❙✉♣❡r✐♦r✿ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♦♥❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❛❜❛✐①♦ ❞❛ ❞✐❛❣♦♥❛❧ sã♦ ♥✉❧♦s✱ ♦✉ s❡❥❛✱ m=n ❡ aij = 0✱ ♣❛r❛ i > j✳

▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ■♥❢❡r✐♦r✿ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♥❛ q✉❛❧ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❛❝✐♠❛ ❞❛ ❞✐❛❣♦♥❛❧ sã♦ ♥✉❧♦s✱ ♦✉ s❡❥❛✱ m =n ❡aij = 0 ♣❛r❛ i < j✳

▼❛tr✐③ ❙✐♠étr✐❝❛✿ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♥❛ q✉❛❧ aij =aji ♣❛r❛ t♦❞♦ i ❡j✳

▼❛tr✐③ ❆♥t✐ss✐♠étr✐❝❛✿ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♥❛ q✉❛❧aij =−aji ♣❛r❛ t♦❞♦i❡j✳

❖♣❡r❛çõ❡s ❝♦♠ ▼❛tr✐③❡s

❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♠❛tr✐③❡s✳

❆ ❆❞✐çã♦ ❡♥tr❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ An×m = [aij] ❡ Bn×m = [bij] é

✉♠❛ ♠❛tr✐③ n×m✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r A+B✱ ❝✉❥♦s ♦s ❡❧❡♠❡♥t♦s sã♦ s♦♠❛s ❞♦s

❡❧❡♠❡♥t♦s ❞❡ A ❡ B✳ ❖✉ s❡❥❛✱

A+B= [aij+bij]_n×m

Pr♦♣r✐❡❞❛❞❡s✿ ❉❛❞❛s ❛s ♠❛tr✐③❡s A✱B✱ C❡ 0 ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ t❡♠♦s✿

❼ A+B=B+C✭❝♦♠✉t❛t✐✈✐❞❛❞❡✮

❼ A+ (B+C) = (A+B) +C ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡✮

▼❛tr✐③❡s ✶✶

▼✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✿ ❙❡❥❛ A= [aij]_n×m ❡ α ✉♠ ♥ú♠❡r♦ r❡❛❧ ✭♦✉ ❝♦♠♣❧❡①♦✮✱

❡♥tã♦ ❞❡✜♥✐♠♦s α·A ❝♦♠♦ ✉♠❛ ♥♦✈❛ ♠❛tr✐③ t❛❧ q✉❡✿

α·A= [α·aij]_m×n

Pr♦♣r✐❡❞❛❞❡s✿ ❉❛❞❛s ♠❛tr✐③❡s A ❡ B ❞❡ ♠❡s♠❛ ♦r❞❡♠(n×m)❡ α✱ β ∈R✭♦✉ C✮

q✉❛✐sq✉❡r✿

❼ α(A+B) = αA+αB

❼ (α+β)A=αA+βA

❼ 0A=0

❼ α(βA) = (αβ)A

❚r❛♥s♣♦s✐çã♦✿ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ A = [aij]_m×n✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ♦✉tr❛ ♠❛tr✐③

AT _{= [b}

ij]_n×m✱ ♥❛ q✉❛❧ ❛s ❧✐♥❤❛s sã♦ ❛s ❝♦❧✉♥❛s ❞❡A✱ ♦✉ s❡❥❛ bij =aji✳ ❆ ♠❛tr✐③ AT é

❝❤❛♠❛❞❛ ❞❡ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ A✳ ❆❧❣✉♥s t❡①t♦s t❛♠❜é♠ ✉s❛♠ ❛ ♥♦t❛çã♦ A′ ♣❛r❛

✐♥❞✐❝❛r ❛ tr❛♥s♣♦st❛ ❞❡ A✳

Pr♦♣r✐❡❞❛❞❡s✿

❼ ❯♠❛ ♠❛tr✐③ é s✐♠étr✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ é ✐❣✉❛❧ ❛ s✉❛ tr❛♥s♣♦st❛ (A=AT₎

❼ (AT₎T _{=A✳ ❆ tr❛♥s♣♦st❛ ❞❛ tr❛♥s♣♦st❛ ❞❡ ✉♠❛ ♠❛tr✐③ é ❡❧❛ ♠❡s♠❛✳}

❼ (A+B)T _=AT_+BT✳ ❆ tr❛♥s♣♦st❛ ❞❡ ✉♠❛ s♦♠❛ é ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s tr❛♥s♣♦st❛s✳

❼ (αA)T _=αAT✱ ♦♥❞❡ _α é ✉♠ ❡s❝❛❧❛r q✉❛❧q✉❡r✳

❆ s❡❣✉✐r ❞❡✜♥✐r❡♠♦s ❛ ♦♣❡r❛çã♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❡♥✈♦❧✈❡♥❞♦ ♠❛tr✐③❡s✿ ❛ ▼✉❧t✐✲ ♣❧✐❝❛çã♦ ❞❡ ▼❛tr✐③❡s✳

❙❡❥❛♠ A= [aij]_n×m ❡ B= [brs]m×p✳ ❉❡✜♥✐♠♦sAB=C= [cuv]n×p ♦♥❞❡

cuv= m

X

k=1

aukbkv =au1b1v+au2b2v +· · ·+aumbmv.

➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ só ♣♦❞❡♠♦s ❡❢❡t✉❛r ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ♠❛tr✐③❡sAn×m❡Bl×p

s❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♣r✐♠❡✐r❛ ❢♦r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛✱ ♦✉ s❡❥❛✱ s❡m=l✳ ❚❛♠❜é♠ ❝❤❛♠❛♠♦s ❛ ❛t❡♥çã♦ ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ♠❛tr✐③ r❡s✉❧t❛❞♦C=AB

s❡rá ❞❡ ♦r❞❡♠ n×p✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❡❧❡♠❡♥t♦ cij ✭✐✲és✐♠❛ ❧✐♥❤❛ ❡ ❥✲és✐♠❛ ❝♦❧✉♥❛✮ é

▼❛tr✐③❡s ✶✷

❆ ✜❣✉r❛ ❛❜❛✐①♦ ✐❧✉str❛ ♦ ♣r♦❞✉t♦ AB✳

a11 . . . a1k . . . a1m

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ai1 . . . aik . . . aim

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ an1 . . . ank . . . anm

                                       

A✿ n❧✐♥❤❛sm❝♦❧✉♥❛s

b11 . . . b1j . . . b1p

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ bk1 . . . bkj . . . bkp

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ bm1 . . . bmj . . . bmp

                                       

B ✿ _m❧✐♥❤❛s _p❝♦❧✉♥❛s

c11 . . . c1j . . . c1p

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ci1 . . . cij . . . cip

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ cn1 . . . cnk . . . cnp

                                       

C=AB✿ _n❧✐♥❤❛s_p❝♦❧✉♥❛s ai1

×b

1j

aik ×bk

j

aim

×bm

j + ✳✳✳+ + ✳✳✳₊ Pr♦♣r✐❡❞❛❞❡s✿

❼ ❊♠ ❣❡r❛❧AB6=BA✭♣♦✐s ✉♠ ♣♦❞❡ ❡st❛r ❞❡✜♥✐❞♦ ❡ ♦ ♦✉tr♦ ♥ã♦✮

❼ AI=IA=A

❼ A(B+C) =AB+AC ✭❞✐str✐❜✉t✐✈❛ à ❡sq✉❡r❞❛✮

❼ (A+B)C=AC+BC ✭❞✐str✐❜✉t✐✈❛ à ❞✐r❡✐t❛✮

❼ (AB)C=A(BC)✭❛ss♦❝✐❛t✐✈✐❞❛❞❡✮ ❼ (AB)T _=BT_AT

❼ 0A=0 ❡ A0=0

❉❡t❡r♠✐♥❛♥t❡✿ é ✉♠ ♥ú♠❡r♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛A = [aij]❡ ❡s❝r❡✈❡✲

♠♦s detA✱ ♦✉ |A| ♦✉ det [aij]✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

det [aij] =

X

ρ

▼❛tr✐③❡s ✶✸

♦♥❞❡ ρ∈Sn é ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ n ❡❧❡♠❡♥t♦s ❡sgnρ= (−1)k✱ ♦♥❞❡k é ♦ ♥ú♠❡r♦ ❞❡

✐♥✈❡rsõ❡s ✭♦✉ tr❛♥s♣♦s✐çõ❡s✮ ❞❡ ρ✳ P♦rt❛♥t♦✱ ❛ s♦♠❛ ❛❝✐♠❛ ❝♦♥té♠ n! ♣❛r❝❡❧❛s✳

❆s ♣r♦♣r✐❡❞❛❞❡s ❛❜❛✐①♦ s♦❜r❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ♦✉ ❞❡ s✉❛ ✐♥✈❡rs❛✱ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✷❪✳

Pr♦♣r✐❡❞❛❞❡s✿

❼ ❙❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ❞❡ ✉♠❛ ♠❛tr✐③Asã♦ ♥✉❧♦s ❡♥tã♦

detA=0✳

❼ detA= detAT

❼ ❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ✉♠❛ ❧✐♥❤❛ ❞❛ ♠❛tr✐③ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ✜❝❛ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ❡st❛ ❝♦♥st❛♥t❡✳

❼ ❯♠❛ ✈❡③ tr♦❝❛❞❛ ❛ ♣♦s✐çã♦ ❞❡ ❞✉❛s ❧✐♥❤❛s✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ tr♦❝❛ ❞❡ s✐♥❛❧ ❼ ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❡ t❡♠ ❞✉❛s ❧✐♥❤❛s ✭❝♦❧✉♥❛s✮ ✐❣✉❛✐s é ③❡r♦ ❼ det(AB) = detAdetB

❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A ❞❡ ♦r❞❡♠ n✱ ❝❤❛♠❛♠♦s ❞❡ ✐♥✈❡rs❛ ❞❡

A ❛ ✉♠❛ ♠❛tr✐③ B t❛❧ q✉❡ AB=BA=In✳ ❯s❛♠♦s A−1 ♣❛r❛ ❛ ✐♥✈❡rs❛ ❞❡ A✳

❼ ❙❡ A ❡ B sã♦ ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ❛♠❜❛s ✐♥✈❡rsí✈❡✐s ✭✐st♦ é✱

❡①✐st❡♠ A−1 _{❡ B}−1_{✮✱ ❡♥tã♦ ABé ✐♥✈❡rsí✈❡❧ ❡ (AB)}−1 _=B−1_A−1_✳

❼ ❙❡ A é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ B t❛❧ q✉❡ BA=I ❡♥tã♦ A é

✐♥✈❡rsí✈❡❧✱ ♦✉ s❡❥❛✱ A−1 _{❡①✐st❡ ❡✱ ❛❧é♠ ❞✐ss♦✱ B=A}−1_✳

❼ ◆❡♠ t♦❞❛ ♠❛tr✐③ t❡♠ ✐♥✈❡rs❛✳

❯♠❛ ✐♥t❡r❡ss❛♥t❡ ❢♦r♠❛ ❞❡ s❛❜❡r s❡ ✉♠❛ ♠❛tr✐③ ♣♦ss✉✐✱ ♦✉ ♥ã♦✱ ✐♥✈❡rs❛ é ❛ ♣❛rt✐r ❞♦ ❝á❧❝✉❧♦ ❞❡ s❡✉ ❞❡t❡r♠✐♥❛♥t❡✱ ♦✉ s❡❥❛✱ s✉♣♦♥❤❛ q✉❡ An×n t❡♥❤❛ ✐♥✈❡rs❛✱ ✐st♦ é✱

❡①✐st❡ A−1 t❛❧ q✉❡_AA−1 _=I

n✳ ❯s❛♥❞♦ ✉♠❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞❡t❡r♠✐♥❛♥t❡ t❡♠♦s✿

det(AA−1_{) = detAdetA}−1 ❡ _detI

n = 1✳ ▲♦❣♦ detAdetA−1 = 1✳ ❉❡ss❡ ♣r♦❞✉t♦

♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ A t❡♠ ✐♥✈❡rs❛ ❡♥tã♦✿

❼ detA6= 0 ❼ detA−1 ₌ 1

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✹

✷✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s

❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ é ✉♠ ❝♦♥❥✉♥t♦ V✱ ♥ã♦ ✈❛③✐♦✱ ❝♦♠ ❞✉❛s ♦♣❡r❛çõ❡s✿ + :V ×V −→V ✭s♦♠❛✮ ❡ ·:R_{×V −→V} ✭♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✮ t❛❧

q✉❡ ♣❛r❛ q✉❛✐sq✉❡r u, v, w ∈V ❡ α, β ∈R ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r sã♦ s❛t✐s❢❡✐t❛s✳

❼ (u+v) +w=u+ (v+w) ❼ u+v =v+u

❼ ❊①✐st❡ 0∈V t❛❧ q✉❡ u+ 0 =u ✭❡❧❡♠❡♥t♦ ♥✉❧♦✮✳ ❼ ❊①✐st❡ −u∈V t❛❧ q✉❡ u+ (−u) = 0

❼ α(u+v) =αu+αv ❼ (α+β)v =αv+βv ❼ (αβ)v =α(βv) ❼ 1u=u

❊①❡♠♣❧♦ ✷✳✹✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❝♦♠ s♦♠❛ ❡ ♣r♦❞✉t♦ ✉s✉❛✐s é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ R✳

❊①❡♠♣❧♦ ✷✳✺✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠2✱ ❝♦♠ ❡♥tr❛❞❛s r❡❛✐s✱ ❞❡✲ ♥♦t❛❞♦ ♣♦rM₂₍R_{) =}

(

a b c d

!

:a, b, c, d∈R

)

❝♦♠ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✱ ❢♦r♠❛ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

❉❡♥tr❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ♣♦ssí✈❡✐s ❞❡ ✉♠ ❡s♣❛ç♦ ❛❧❣✉♥s s❡ ❞❡st❛❝❛♠ ♣♦r ❛❧❣✉✲ ♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❊ss❡s s✉❜❝♦♥❥✉♥t♦s ♠♦t✐✈❛♠ ❛ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✳

❉❡✜♥✐çã♦ ✷✳✻✳ ❉❛❞♦ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ W✱ ♥ã♦ ✈❛③✐♦✱ s❡rá ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ V s❡✿

❼ P❛r❛ q✉❛✐sq✉❡r u, v ∈W ❡♥tã♦ u+v ∈W ❼ P❛r❛ q✉❛✐sq✉❡r α∈R_{, u∈W} ❡♥tã♦ _{αu ∈W}

❱❛❧❡ ❛ ♣❡♥❛ ❞❡st❛❝❛r q✉❡ ♣❛r❛ q✉❡W s❡❥❛ ✉♠ s✉❜❡s♣❛ç♦ ❞❡ ✈❡t♦r✐❛❧ ❞❡V ❡❧❡ ❞❡✈❡ ❝♦♥t❡r ♦ ❡❧❡♠❡♥t♦ ♥✉❧♦ ❞❡ V✱ ❛❧✐ás✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛♥❞♦ ❛♣❡♥❛s ❡❧❡♠❡♥t♦ ♥✉❧♦ ❞❡ V é ✉♠ s✉❜❡s♣❛ç♦✳

❊①❡♠♣❧♦ ✷✳✼✳ ❙❡ V = M₂₍R₎ ❡ _W=

(

α β 0 0

!

:α, β ∈R

)

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✺

❉❡✜♥✐çã♦ ✷✳✽✳ ❙❡❥❛♠ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ✭♦✉ ❝♦♠♣❧❡①♦✮✱ v1, v2, . . . , vn ∈ V ❡

α1, . . . , αn ♥ú♠❡r♦s r❡❛✐s ✭♦✉ ❝♦♠♣❧❡①♦s✮✳ ❊♥tã♦ ♦ ✈❡t♦r

v =α1v1+α2v2+· · ·+αnvn

é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ v1,· · · , vn

❉❡✜♥✐çã♦ ✷✳✾✳ ❋✐①❛❞♦s v1, v2,· · · , vn ∈V ❝❤❛♠❛♠♦s ❞❡ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ❛♦ ❝♦♥✲

❥✉♥t♦

W ={v ∈V;v =α1v1+α2v2+· · ·+αnvn, ai ∈R,1≤i≤n}.

◆♦t❛çã♦✿ W = [v1, v2,· · · , vn].

◆♦ ❊①❡♠♣❧♦ ✷✳✼ ♦s ❡❧❡♠❡♥t♦s v1 =

" 1 0 0 0 #

❡ v2 =

" 0 1 0 0 #

❣❡r❛♠ ♦ s✉❜❡s♣❛ç♦ W✳

P♦rt❛♥t♦ W = [v1, v2] =

("

α β 0 0 #

:α, β ∈R

)

✳

❉❡✜♥✐çã♦ ✷✳✶✵✳ ❙❡❥❛♠V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡v1, . . . , vn∈V✳ ❉✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦

{v1, . . . , vn} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ✭▲■✮ s❡

a1v1+· · ·+anvn = 0

s♦♠❡♥t❡ q✉❛♥❞♦ a1 = a2 = · · · = an = 0✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ é

❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s ✭▲❉✮✳

❉❡✜♥✐çã♦ ✷✳✶✶✳ ❯♠ ❝♦♥❥✉♥t♦ {v1, . . . , vn} ❞❡ ✈❡t♦r❡s ❞❡ V s❡rá ✉♠❛ ❜❛s❡ ❞❡ V ✭❡

♥❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ V t❡♠ ❜❛s❡ ✜♥✐t❛✮ s❡✿ ❼ {v1, . . . , vn} é ▲■

❼ [v1, . . . , vn] =V

❊①❡♠♣❧♦ ✷✳✶✷✳ ❙❡❥❛ V = R2 ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ _{v1 = (1,1)} ❡ _{v2 = (0,1)}✳ ▼♦str❛✲

r❡♠♦s q✉❡ v1 ❡ v2 é ✉♠❛ ❜❛s❡ ❞❡R2✳ ❙❡(0,0) = α(1,1) +β(0,1) = (α, α+β)✱ ❡♥tã♦

α =β = 0✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ v1 ❡ v2 sã♦ ▲■✳

❚❛♠❜é♠ t❡♠♦s q✉❡[(1,1),(0,1)] =R2✱ ♣♦✐s ❞❛❞♦ q✉❛❧q✉❡r_{v = (x, y)∈}R2♣♦❞❡♠♦s

❡♥❝♦♥tr❛r α ❡ β r❡❛✐s t❛✐s q✉❡

(x, y) =α(1,1) +β(0,1)

♥❡st❡ ❝❛s♦ ❜❛st❛ t♦♠❛r α=x ❡β =y−x✳

❚❡♦r❡♠❛ ✷✳✶✸✳ ❙❡❥❛♠ v1, v2, . . . , vn ✈❡t♦r❡s ♥ã♦ ♥✉❧♦s q✉❡ ❣❡r❛♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✻

❉❡♠♦♥str❛çã♦✳ ❙❡ v1, v2, . . . , vn sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tã♦ ♥ã♦ ❤á ♥❛❞❛ ❛

♠♦str❛r✳ ❙❡ v1, v2, . . . , vn sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s ❡♥tã♦ ❡①✐st❡ ❛❧❣✉♠ ❝♦❡✜❝✐❡♥t❡

❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ t❛❧ q✉❡

x1v1 +x2v2+· · ·+xnvn= 0.

❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡ xn6= 0✳ ❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

vn =

−x1

xn

v1+

−x2

xn

v2+· · ·

−xn−1

xn

vn−1

♦✉ s❡❥❛✱vné ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡v1, . . . , vn−1❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱v1, v2, . . . , vn−1

❛✐♥❞❛ ❣❡r❛♠ V✳ ❙❡ v1, v2, . . . vn−1 ❢♦r ▲❉✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡❧❡s

✐❣✉❛❧ ❛♦ ✈❡t♦r ♥✉❧♦ ❝♦♠ ❛❧❣✉♠ ❝♦❡✜❝✐❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦❀ ❞❡ss❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ❡①✲ tr❛✐r ♦ ✈❡t♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡ss❡ ❝♦❡✜❝✐❡♥t❡✳ Pr♦❝❡❞❡♥❞♦ ❞❡st❛ ❢♦r♠❛✱ ❛♣ós ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♣❛ss♦s✱ ❝❤❡❣❛r❡♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ {v1, . . . vn}✱ ❢♦r♠❛❞♦ ♣♦r

r (r ≤n) ✈❡t♦r❡s ▲■ q✉❡ ❣❡r❛♠ V✱ ✐st♦ é✱ t❡r❡♠♦s ✉♠❛ ❜❛s❡✳

❚❡♦r❡♠❛ ✷✳✶✹✳ ❙❡❥❛ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ❣❡r❛❞♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ✈❡t♦r❡s v1, v2, . . . , vn✳ ❊♥tã♦ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♠ ♠❛✐s ❞❡ n ✈❡t♦r❡s é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ▲❉

✭❡✱ ♣♦rt❛♥t♦ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ▲■ t❡♠ ♥♦ ♠á①✐♠♦ n ✈❡t♦r❡s✮✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ [v1, . . . , vn] = V ♣❡❧♦ t❡♦r❡♠❛ ✷✳✶✸ ♣♦❞❡♠♦s ❡①tr❛✐r ✉♠❛ ❜❛s❡

♣❛r❛V ❞❡v1, . . . , vn✳ ❙❡❥❛v1, . . . vr✱r≤n✱ ❡st❛ ❜❛s❡✳ ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛w1, w2, . . . , wm✱

m ✈❡t♦r❡s ❞❡ V✱ ❝♦♠m > n✳ ▲♦❣♦✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s aij✱ t❛✐s q✉❡

w1 = a11v1+a12v2+· · ·+a1rvr

w2 = a21v1+a22v2+· · ·+a2rvr

✳✳✳ ✳✳✳

wm = am1v1+am2v2+· · ·+amrvr

✭✷✳✶✮

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ w1, . . . , wm

x1w1+x2w2+· · ·+xmwm = 0. ✭✷✳✷✮

❙✉❜st✐t✉✐♥❞♦ ❛s r❡❧❛çõ❡s ✭✷✳✶✮ ❡♠ ✭✷✳✷✮ ❡ ❢❛③❡♥❞♦ ♦s ❛❣r✉♣❛♠❡♥t♦s ♥❡❝❡ssár✐♦s t❡♠♦s✿

0 = (a11x1+a21x2 +· · ·+am1xm)v1+ (a12x1+a22x2+· · ·+am2xm)v2+· · ·

· · ·+ (a1rx1+a2rx2+· · ·+amrxm)vr.

❈♦♠♦v1, v2, . . . , vr sã♦ ▲■✱ ❡♥tã♦

     

    

a11x1+a21x2+· · ·+am1xm = 0

a12x1+a22x2+· · ·+am2xm = 0

✳✳✳

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✼

❚❡♠♦s ❡♥tã♦ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡♦ ❝♦♠r❡q✉❛çõ❡s ❡m✐♥❝ó❣♥✐t❛sx1, . . . , xm

❡✱ ❝♦♠♦ r ≤ n ≤ m ❡❧❡ ❛❞♠✐t❡ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❝♦♠ ❛❧❣✉♠ xi ♥ã♦ ♥✉❧♦✳ P♦rt❛♥t♦ w1, . . . , wm sã♦ ▲❉✳

❈♦r♦❧ár✐♦ ✷✳✶✺✳ ◗✉❛❧q✉❡r ❜❛s❡ ❞❡ ✉♠ ♠❡s♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t❡♠ s❡♠♣r❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✱ ♦✉ s❡❥❛✱ ❞✐♠❡♥sõ❡s ✐❣✉❛✐s✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ {v1, . . . , vn} ❡ {w1, . . . , wm} ❞✉❛s ❜❛s❡s ❞❡ V✳ ❯♠❛ ✈❡③ q✉❡

v1, . . . , vn ❣❡r❛♠ ❱ ❡ w1, . . . , wm sã♦ ▲■✱ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ m≤n.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ w1, . . . , wm ❣❡r❛♠ V ❡ v1, . . . , vn sã♦ ▲■✱ ❛✐♥❞❛ ♣❡❧♦ t❡♦r❡♠❛

✷✳✶✹✱ n ≤m✳ P♦rt❛♥t♦ n=m✳

❉❡✜♥✐çã♦ ✷✳✶✻✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣♦ss✉✐♥❞♦ ✉♠❛ ❜❛s❡ ✜♥✐t❛✳ ❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡st❛ ❜❛s❡ ✭❡ ♣♦rt❛♥t♦✱ ❞❡ q✉❛❧q✉❡r ♦✉tr❛✮ é ❝❤❛♠❛❞♦ ❞❡ ❞✐♠❡♥sã♦ ❞❡ V ❡ ❞❡♥♦t❛❞♦ ♣♦r dimV✳ ❙❡ V ={0} ❝♦♥✈❡♥❝✐♦♥❛✲s❡ dimV = 0✳

❉❡✜♥✐çã♦ ✷✳✶✼ ✭❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r✮✳ ❙❡❥❛♠ V ❡ W ❞♦✐s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r é ✉♠❛ ❢✉♥çã♦ ❞❡ V ❡♠ W✱ F : V → W✱ q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✐✮ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ u, v∈V✱

F(u+v) = F(u) +F(v)

✐✐✮ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ α∈R ❡ _{v ∈V}✱

F(αv) = αF(v)

❯♠ ✐♠♣♦rt❛♥t❡ ❡①❡♠♣❧♦ é q✉❡ t♦❞❛ ♠❛tr✐③n×m ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ tr❛♥s❢♦r♠❛✲ çã♦ ❧✐♥❡❛r ❞❡ Rm ❡♠ _Rn✳ P♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠❛ ♠❛tr✐③ ♣r♦❞✉③ ✉♠❛ tr❛♥s❢♦r♠❛çã♦

❧✐♥❡❛r✳ ❆ ✐♠♣❧✐❝❛çã♦ ✐♥✈❡rs❛ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣♦✐s✱ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡

Rm ❡♠ _Rn ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ♠❛tr✐③ _n×m✳ ❆ s❛❜❡r s❡❥❛ _A ✉♠❛ ♠❛tr✐③

n×m✳ ❉❡✜♥✐♠♦s

LA : Rm −→ Rn

v 7−→ A·v

♦♥❞❡ v ∈Rm_{✱ v =}



 

x1

✳✳✳ xm



 

LA(v) = A· 

 

x1

✳✳✳ xm





=



 

y1

✳✳✳ yn



 

❉❛❞♦su, v ∈Rm ❡_{α ∈R} ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s s❡❣✉❡ q✉❡✿

LA(u+v) = A(u+v) =Au+Av =LA(u) +LA(v)❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦

❞❡ ✉♠❛ ♠❛tr✐③ ♣♦r ✉♠ ❡s❝❛❧❛r t❡♠♦s✿ LA(αv) = A(αv) =αA(v) =αLA(v)❡ ♣♦rt❛♥t♦

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✽

■♠❛❣❡♠ ❡ ◆ú❝❧❡♦✳ ❙❡❥❛ T : V → W ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❆ ✐♠❛❣❡♠ ❞❡ T é ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s w∈W t❛✐s q✉❡ ❡①✐st❡ ✉♠ ✈❡t♦r v ∈ V✱ q✉❡ s❛t✐s❢❛③ T(v) =w✳ ❖✉ s❡❥❛✱

im(T) ={w∈W; T(v) =w ♣❛r❛ ❛❧❣✉♠ v ∈V}.

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✈❡t♦r❡sv ∈V t❛✐s q✉❡T(v) = 0é ❝❤❛♠❛❞♦ ❞❡ ♥ú❝❧❡♦ ❞❡ T✱ s❡♥❞♦ ❞❡♥♦t❛❞♦ ♣♦r ker(T)✳ ■st♦ é

ker(T) = {v ∈V; T(v) = 0}.

❱❛❧❡ r❡ss❛❧t❛r q✉❡ t❛♥t♦ im(T) ⊂ W q✉❛♥t♦ ker(T) ⊂ V sã♦ s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❈❤❛♠❛♠♦s ❞❡ ♣♦st♦ (T)✱ ❞❡♥♦t❛❞♦ ♣♦rrkT✱ ❛ ❞✐♠❡♥sã♦ ❞❛ ✐♠❛❣❡♠ ❞❡ T✳

❚❡♦r❡♠❛ ✷✳✶✽✳ ❙❡ T : V → W é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❡♥tã♦ ker(T) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ T é ✐♥❥❡t♦r❛✳

❉❡♠♦♥str❛çã♦✳ ✭⇒✮ ❙✉♣♦♥❤❛ q✉❡u, v ∈V t❛✐s q✉❡T(u) = T(v)✳ ❊♥tã♦T(u)−T(v) = T(u−v) = 0✱ ♦✉ s❡❥❛✱ u−v ∈ker(T)✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ ♥ú❝❧❡♦ é 0✱ ❡♥tã♦ u−v = 0✱ ♦✉ s❡❥❛✱ u=v✳

✭⇐✮ ❙❡❥❛ v ∈ ker(T)✱ ✐st♦ é✱ T(v) = 0✳ ❈♦♠♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ T(0) = 0✱ T(v) = T(0)✳ ▲♦❣♦ v = 0✱ ♣♦✐s T é ✐♥❥❡t♦r❛✳ P♦rt❛♥t♦ ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ ♥ú❝❧❡♦ é0✱ ♦✉ s❡❥❛✱ ker(T) = {0}

❆❣♦r❛ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛s ❞✐♠❡♥sõ❡s ❞♦ ♥ú✲ ❝❧❡♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳

❚❡♦r❡♠❛ ✷✳✶✾ ✭❞♦ ◆ú❝❧❡♦ ❡ ❞❛ ■♠❛❣❡♠✮✳ ❙❡❥❛ V ❡ W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ T :V →W ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❊♥tã♦

dim ker(T) + dim im(T) = dimV.

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ v1, . . . , vn ✉♠❛ ❜❛s❡ ❞❡ ker(T)✳ ❈♦♠♦ ker(T)⊂ V é s✉❜❡s✲

♣❛ç♦ ❞❡ V✱ ♣♦❞❡♠♦s ❝♦♠♣❧❡t❛r ❡st❡ ❝♦♥❥✉♥t♦ ❞❡ ♠♦❞♦ ❛ ♦❜t❡r ✉♠❛ ❜❛s❡ ❞❡ V✳

❙❡❥❛ ❡♥tã♦{v1, . . . , vn, w1, . . . , wm}❛ ❜❛s❡ ❞❡V✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡T(w1), . . . , T(wm)

é ✉♠❛ ❜❛s❡ ❞❡ im(T)✱ ✐st♦ é✱ ✐✮ [T(w1), . . . , T(wm)] = im(T)

✐✐✮ {[T(w1), . . . , T(wm)} é ▲■

Pr♦✈❛ ❞❡ ✐✮✿ ❉❛❞♦ w ∈ im(T) ❡①✐st❡ u ∈ V t❛❧ q✉❡ T(u) = w✳ ❙❡ u ∈ V✱ ❡♥tã♦ u=a1v1+· · ·+anvn+b1w1+· · ·+bmwm✳ ▼❛s✱

w =T(u) = T(a1v1+· · ·+anvn+b1w1+· · ·+bmwm)

=a1T(v1) +· · ·+anT(vn) +b1T(w1) +· · ·+bmT(wm)

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✶✾

❈♦♠♦ v1, . . . , vn ∈ker(T)✱T(vi) = 0 ♣❛r❛ i= 1, . . . , n✳ ❉❡ss❡ ♠♦❞♦✱

w=b1T(w1) +· · ·+bmT(wm)

❡ ❛ ✐♠❛❣❡♠ ❞❡ T é ❣❡r❛❞❛ ♣❡❧♦s ✈❡t♦r❡s T(w1), . . . , T(wm)✳

Pr♦✈❛ ❞❡ ✐✐✮✿ ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r α1T(w1) + α2T(w2) +· · · +

αmT(wm) = 0.▼♦str❛r❡♠♦s q✉❡ t♦❞♦s ♦s αi sã♦ ♥✉❧♦s✳

❈♦♠♦T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r T(α1w1+α2w2+· · ·+αmwm) = 0✳ P♦rt❛♥t♦

α1w1+α2w2 +· · ·+αmwm ∈ ker(T)✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ α1w1+α2w2+· · ·+αmwm

♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛ ❜❛s❡ {v1, . . . , vn} ❞❡ ker(T)✱ ✐st♦ é✱

❡①✐st❡♠ β1, . . . , βn t❛✐s q✉❡

α1w1+α2w2+· · ·+αmwm =β1v1+· · ·+βnvn

α1w1+α2w2+· · ·+αmwm−β1v1− · · · −βnvn= 0

.

▼❛s {v1, . . . , vn, w1, . . . , wm} é ✉♠❛ ❜❛s❡ ❞❡ V ❡♥tã♦ t❡♠♦s α1 = α2 = · · · = αm =

β1 =· · ·=βn = 0✳

◆♦t❡ q✉❡ dim ker(T) =n✱ dim im(T) =m ❡dimV =n+m✳

❙✉❜❡s♣❛ç♦s ❋✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ ▼❛tr✐③✳ ❋✐♥❛❧✐③❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❞❡✜♥✐♥❞♦ ❞♦✐s s✉❜❡s♣❛ç♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ ♠❛tr✐③ ❡ ♦s r❡❧❛❝✐♦♥❛♥❞♦ ❝♦♠ ♦ t❡♦r❡♠❛ ❞♦ ♥ú❝❧❡♦ ❡ ✐♠❛❣❡♠✳ ❊ss❡s ❝♦♥❝❡✐t♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ❛s ❞❡✜♥✐çõ❡s ❛ s❡❣✉✐r ❝♦♥s✐❞❡r❡ A ∈ M_n×m(R₎✱ ✐st♦ é✱ _A é ✉♠❛ ♠❛tr✐③ ❞❡

♦r❞❡♠ n×m ❝♦♠ ❡♥tr❛❞❛s r❡❛✐s✳

❊s♣❛ç♦ ❈♦❧✉♥❛ ❞❡ A✿ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ Rn _{❞❡✜♥✐❞♦ ♣♦r}

R(A) ={z ∈Rn _{|z =Ax;x∈}Rm_}.

❖❜s❡r✈❡ q✉❡R(A)é ✉♠ s✉❜❡s♣❛ç♦ ❞❡Rn_{❣❡r❛❞♦ ♣❡❧❛s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③A✳ ❯t✐❧✐③❛♥❞♦}

❛ ♥♦t❛çã♦ A= [c1, . . . , cm] ♣❛r❛ j ∈ {1,2, . . . , m}✱ ♦♥❞❡ cj ∈ Rn é ❛ ❥✲és✐♠❛ ❝♦❧✉♥❛ ❞❛

♠❛tr✐③✱ t❡♠♦s q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ z ∈ R(A) ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿

z =Ax =

m

X

j=1

xjcj ; x=



 

x1

✳✳✳ cm



 ∈Rn.

❊s♣❛ç♦ ◆✉❧♦ ❞❡ A✿ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ Rm _{❞❡✜♥✐❞♦ ♣♦r}

N(A) ={x∈Rm _{|Ax= 0R}n}.

◆♦t❡ q✉❡ N(A) é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ Rm✱ ❝♦♥st✐t✉í❞♦ ♣❡❧❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r

❤♦♠♦❣ê♥❡♦ Ax= 0Rn✳

❚❛♠❜é♠ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ♦ ❡s♣❛ç♦ ❝♦❧✉♥❛ ❞❡ AT ❝♦♠♦ ♦ s✉❜❡s♣❛ç♦ ❞❡_Rm ❣❡r❛❞♦

❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✷✵

♥✉❧♦ ❞❡ AT q✉❡ é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ Rn✳ ❊♠ ❛❧❣✉♠❛s ♦❝❛s✐õ❡s ♦ ❡s♣❛ç♦ ♥✉❧♦ ❞❡ _AT é

❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ♥✉❧♦ ❡sq✉❡r❞♦ ❞❡ A✱ ✐ss♦ s❡ ❞❡✈❡ ❛♦ s❡❣✉✐♥t❡ ❢❛t♦✿ s❡ ♦ ❡❧❡♠❡♥t♦

z ∈ N(AT) ❡♥tã♦ ATz = 0Rm ⇔zTA=0T_Rm✳

❖ s✉❜❡s♣❛ç♦ ❝♦❧✉♥❛ ❡ ♦ s✉❜❡s♣❛ç♦ ♥✉❧♦ ♣♦ss✉❡♠ ✉♠❛ ❢♦rt❡ r❡❧❛çã♦ ❝♦♠✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✱ ♦ s✉❜❡s♣❛ç♦ ✐♠❛❣❡♠ ❡ ♦ s✉❜❡s♣❛ç♦ ♥ú❝❧❡♦ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ■ss♦ s❡ ❞❡✈❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r s❡♠♣r❡ ❡stá✱ ♦✉ ♣♦❞❡ s❡r✱ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ♠❛tr✐③✳ ❆ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦ ❝♦❧✉♥❛ ❞❡ ✉♠❛ ♠❛tr✐③ A t❛♠❜é♠

é ❝❤❛♠❛❞❛ ❞❡ ♣♦st♦ ❞❡ A ❡ t❛♠❜é♠ ❞❡♥♦t❛❞♦ ♣♦r rkA✳

❚❡♥❞♦ ❡♠ ✈✐st❛ ❡ss❡ ❢❛t♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ t❡♦r❡♠❛ ✷✳✶✾ ❡♠ t❡r♠♦s ❞♦ s✉❜❡s♣❛ç♦ ❝♦❧✉♥❛ ❡ s✉❜❡s♣❛ç♦ ♥✉❧♦ ❞❡ ✉♠❛ ♠❛tr✐③✳ ❙❡❥❛ An×m ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ (n ×m)✳

◆❡ss❡ ❝❛s♦ t❡♠♦s✿

✸ ❈♦♥❝❡✐t♦s ❊❧❡♠❡♥t❛r❡s ❞❛ ❚❡♦r✐❛ ❞❡

●r❛❢♦s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐r❡♠♦s ❣r❛❢♦s ❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s✳ P❛r❛ ✐ss♦✱ ❢❛r❡♠♦s ✉s♦ ❞❛ ♥♦çã♦ ❞❡ ❢❛♠í❧✐❛ q✉❡✱ ❛ ♣❛rt✐r ❞❛q✉✐✱ s❡rá ✉♠ ❝♦♥❥✉♥t♦ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❝♦♥❥✉♥t♦s✳

❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠ ❣r❛❢♦ ● é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♣❛r (V(G), E(G)) ♦♥❞❡ V(G) é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ E(G) ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣❛r❡s ♥ã♦ ♦r❞❡♥❛❞♦s ❞❡ ❡❧❡♠❡♥t♦s✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐st✐♥t♦s✱ ❞❡ V(G)✳

❉❡♥♦♠✐♥❛r❡♠♦s ❞❡ ❣r❛❢♦ s✐♠♣❧❡s G✱ ❝♦♠♦ ✉♠ ❣r❛❢♦G♥♦ q✉❛❧ ♥ã♦ ❡①✐st❡ r❡♣❡t✐çõ❡s ♥♦s ❡❧❡♠❡♥t♦s ❞❡ E(G)✱ ❛❧é♠ ❞✐ss♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ E(G) é ✉♠ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❡ ♥ã♦ ♦r❞❡♥❛❞♦s ❞❡ V(G)✳ ❆♦ ❧♦♥❣♦ ❞❡ss❡ tr❛❜❛❧❤♦ ♦ ♥♦ss♦ ❢♦❝♦ ❡st❛rá ❡♠ ❣r❛❢♦s s✐♠♣❧❡s✱ ♥♦s q✉❛✐s ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s é ✜♥✐t♦✳ ◆♦t❡ q✉❡ ❛ ✐♥t❡♥çã♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❣r❛❢♦ é ❛ ❞❡ s❡r ❛ ♠❛✐s ❛❜r❛♥❣❡♥t❡ ♣♦ssí✈❡❧✱ ♠♦str❛♥❞♦ ❛ ❡ssê♥❝✐❛ ❞❛ ❡str✉t✉r❛ ❞❡✜♥✐❞❛ ❝♦♠♦ ❣r❛❢♦✳

❖s ❡❧❡♠❡♥t♦s ❞❡V(G)s❡rã♦ ❝❤❛♠❛❞♦s ❞❡ ✈ért✐❝❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡E(G)❞❡ ❛r❡st❛s✳ ◗✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦ ❞❡♥♦t❛r❡♠♦s V(G) ❡ E(G)s✐♠♣❧❡s♠❡♥t❡ ♣♦r V ❡ E✳ ❯♠❛ ❛r❡st❛ ❞❡ E(G)✱ ♣♦r ❡①❡♠♣❧♦ {a, b} s❡rá ❞❡♥♦t❛❞❛ ♣♦r ab✱ ❡ ♥❡ss❡ ❝❛s♦ ♦s

✈ért✐❝❡sa❡bsã♦ ❛❞❥❛❝❡♥t❡s✳ ❚❛♠❜é♠ ❞✐③❡♠♦s q✉❡ ❞✉❛s ❛r❡st❛s sã♦ ❛❞❥❛❝❡♥t❡s q✉❛♥❞♦ ♣♦ss✉ír❡♠ ✉♠ ✈ért✐❝❡ ❡♠ ❝♦♠✉♠✱ ♣♦r ❡①❡♠♣❧♦ ❞❛❞❛s ab, cd ∈ E(G) ❡♥tã♦ ♦✉ a = b ♦✉ d✱ ♦✉ b =c ♦✉ d✳ ❯♠❛ ❛r❡st❛ é ✐♥❝✐❞❡♥t❡ ❛♦ ✈ért✐❝❡ a q✉❛♥❞♦ ❡❧❡ ❢♦r ✉♠❛ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s✳ ❉❡✜♥✐♠♦s ❝♦♠♦ ❛r❡st❛ ♠ú❧t✐♣❧❛ ✉♠❛ q✉❡ ❛r❡st❛ ❛♣❛r❡❝❡ ♠❛✐s ❞❡ ✉♠❛ ✈❡③ ♥♦ ❣r❛❢♦G❀ ♦ ♥ú♠❡r♦ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡st❛ ❛r❡st❛ é ❝❤❛♠❛❞♦ ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✳ ❯♠❛ ❛r❡st❛ é ✉♠ ❧❛ç♦ s❡ ♣❛r❛ v ∈ V(G), vv ∈ E(G)✳ ❖ ❣r❛✉ ❞❡ ✉♠ ✈ért✐❝❡ v é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥tê♠ v✱ ❞❡♥♦t❛❞♦ ♣♦r g(v)✳ ❈❛s♦ ❡①✐st❛♠ ❧❛ç♦s ✐♥❝✐❞❡♥t❡s ♥♦ ✈ért✐❝❡ v✱ ❝❛❞❛ ❧❛ç♦ ❝♦♥tr✐❜✉✐rá ❡♠ ❞✉❛s ✉♥✐❞❛❞❡s ♣❛r❛ ♦ ❣r❛✉ ❞❡ v✳ ❯♠ ✈ért✐❝❡ í♠♣❛r é ✉♠ ✈ért✐❝❡ ❝♦♠ ❣r❛✉ í♠♣❛r✳

❖ ❣r❛✉ ♠á①✐♠♦ ❞❡ ✉♠ ❣r❛❢♦ G✱ ❞❡♥♦t❛❞♦ ♣♦r ∆(G)✱ é ❞❡✜♥✐❞♦ ♣♦r✿ ∆(G) = max{g(v)|v ∈V(G)}✱ ♦✉ s❡❥❛✱ ❞❡♥tr❡ t♦❞♦s ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦ ❛q✉❡❧❡ q✉❡ ♣♦ss✉✐ ♦

♠❛✐♦r ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ P♦r ♦✉tr♦ ❧❛❞♦ ♦ ❣r❛✉ ♠í♥✐♠♦✱ ❞❡♥♦t❛❞♦ ♣♦rδ(G)é ❞❡✜♥✐❞♦ ❝♦♠♦ δ(G) = min{g(v)|v ∈V(G)}✳

✷✷

▲❡♠❛ ✸✳✷✳ ❙❡❥❛ G ✉♠ ❣r❛❢♦ ❝♦♠ V = {a1, a2, ..., an}✱ ❝✉❥♦s ❣r❛✉s sã♦ ❞❛❞♦s ♣♦r

g(a1), g(a2), ..., g(an)✳ ❖ ♥ú♠❡r♦ m ❞❡ ❛r❡st❛s ❡♠ G é ❞❛❞♦ ♣♦r✿

m =

g(a1) +g(a2) +· · ·+g(an)

2

.

❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞❡ G é ✉♠ ♥ú♠❡r♦ ♣❛r✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❝❛❞❛ ✈ért✐❝❡ ai ❢♦r♥❡❝❡ g(ai) ❡ ❝♦♠♦ ❝❛❞❛ ❛r❡st❛ ❝♦♥té♠ ❡①❛✲

t❛♠❡♥t❡ ❞♦✐s ✈ért✐❝❡s✱ ❞❡✈❡♠♦s ❡♥tã♦ t♦♠❛r ❛ ♠❡t❛❞❡ ❞❛ s♦♠❛ ❞♦s ❣r❛✉s✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ ▲❡♠❛ t❡♠♦s✿

❚❡♦r❡♠❛ ✸✳✸✳ ❚♦❞♦ ❣r❛❢♦ G t❡♠ ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ✈ért✐❝❡s í♠♣❛r❡s

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛G✉♠ ❣r❛❢♦ ❝♦♠V ={w1, . . . , wj, v1, . . . , vn}✱ s✉♣♦♥❤❛ q✉❡g(wi) =

2k′

i=1,...,j ∈N ❡g(vi) = 2ki+ 1 ♣❛r❛ ki=1,...,n ∈N✳ ❊♥tã♦

g(w1) +· · ·+g(wj) +g(v1)· · ·+g(vn) =

(2k′

1) +· · ·+ (2kj′) + (2k1+ 1) +· · ·+ (2kn+ 1) =

2·(k′

1+· · ·+kj′ +k1+· · ·+kn) +n

▼❛s✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❛ s♦♠❛ ❞❡✈❡ s❡r ✉♠ ♥ú♠❡r♦ ♣❛r✱ ♣♦rt❛♥t♦ n ✭q✉❡ é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r✮ ❞❡✈❡ s❡r ✉♠ ♥ú♠❡r♦ ♣❛r✳

❆ ♦r❞❡♠ ❞❡ ✉♠ ❣r❛❢♦ G é ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ V✱ ❞❡♥♦t❛❞♦ ♣♦r |V|✱ ❡ ❛

❞✐♠❡♥sã♦ ❞❡ G é ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦E✱ ❞❡♥♦t❛❞♦ ♣♦r |E|✳ ❉✐③❡♠♦s q✉❡ ✉♠

❣r❛❢♦ H é s✉❜❣r❛❢♦ ❞❡ G s❡V(H)⊆V(G)❡ E(H)⊆E(G)✳

❈♦♥s✐❞❡r❡ ♦ ❣r❛❢♦ G = (V(G), E(G)) ❝♦♠ s✉❜❣r❛❢♦s H1, . . . , Hp ♦♥❞❡✿ V(Hi)∩

V(Hj) =∅ ♣❛r❛ i6=j, V(H1)∪ · · · ∪V(Hp) =V(G) ❡ E(H1)∪ · · · ∪E(Hp) =E(G).

❖s s✉❜❣r❛❢♦s ♥❡ss❡ ❝❛s♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ G✱ ❡ ❛ss✐♠ ❞✐③❡♠♦s q✉❡ G t❡♠ p ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✳

❯♠ ❣r❛❢♦ é ❞✐t♦ ❝♦♥❡①♦ q✉❛♥❞♦ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ é ❞✐t♦ ❞❡s❝♦♥❡①♦✳ ❱❛❧❡ ♦❜s❡r✈❛r q✉❡✱ ❡♠ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✱ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❝♦♥❡❝t❛r ❞♦✐s ✈ért✐❝❡s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛r❡st❛s ❛❞❥❛❝❡♥t❡s✳

❯♠ ❣r❛❢♦ ❝♦♠ n ✈ért✐❝❡s s❡rá r❡❣✉❧❛r ❞❡ ❣r❛✉ ❦ ♦✉ ❦✲r❡❣✉❧❛r✱ q✉❛♥❞♦ ♦ ❣r❛✉ ❞❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ✈ért✐❝❡s ❢♦r ✐❣✉❛❧ ❛ k✳ P♦❞❡✲s❡ ✈❡r✐✜❝❛r ❢❛❝✐❧♠❡♥t❡ ♣❡❧♦ ▲❡♠❛ ✸✳✷ q✉❡ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s s❡rá m = 1₂nk✳ ❯♠ ❣r❛❢♦ ❝✐❝❧♦✱ ❞❡ n ✈ért✐❝❡s ❡ ❞❡♥♦t❛❞♦ ♣♦r Cn é

✉♠ ❣r❛❢♦ 2✲r❡❣✉❧❛r ❝♦♥❡①♦✳

❯♠ ❣r❛❢♦Gs❡♠ ❧❛ç♦s ❡ s❡♠ ❛r❡st❛s ♠ú❧t✐♣❧❛s é ❝♦♠♣❧❡t♦ s❡ ♣❛r❛ q✉❛✐sq✉❡ra, b∈V t❡♠♦s ab ∈ E✳ ❯♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❝♦♠ n ✈ért✐❝❡s é ❞❡♥♦t❛❞♦ ♣♦r Kn ❡ ❞✐③❡♠♦s q✉❡

✉♠ ❣r❛❢♦ é ♥✉❧♦ q✉❛♥❞♦ E(G) =∅✱ ❡ ♦ ❞❡♥♦t❛♠♦s ♣♦r Kn✳

❉♦✐s ❣r❛❢♦s H1 = (V(H), E(H1)) ❡ H2 = (V(H), E(H2)) ✭❝♦♠ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦

✷✸

✷✳ ♦ ❣r❛❢♦H = (V(H), E(H1)∪E(H2))é ❝♦♠♣❧❡t♦✳

❉❡♥♦t❛♠♦s ♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ♣♦r H1 =H2 ✭❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ H2 =H1✮✳

❈♦r♦❧ár✐♦ ✸✳✹✳ ❖ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❡♠ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ❝♦♠♣❧❡t♦ Kn é n(n₂−1)✳

❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ❝♦♥s✐❞❡r❛r ♦ ❢❛t♦ q✉❡ Kn é ✉♠ ❣r❛❢♦ r❡❣✉❧❛r ❞❡ ❣r❛✉ (n −1)✳

P❡❧♦ ▲❡♠❛ ✸✳✷ t❡♠♦s q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s s❡rá n(n−1) 2 .

❯♠❛ ❞❡♠♦♥str❛çã♦ ❛❧t❡r♥❛t✐✈❛ ♣♦❞❡ s❡r ♣♦r ✐♥❞✉çã♦ ❡♠n✳ P❛r❛ n = 1 ♦ r❡s✉❧t❛❞♦ é ✐♠❡❞✐❛t♦✳ ❙✉♣♦♥❤❛ ✈á❧✐❞♦ ♣❛r❛ ✉♠ ❣r❛❢♦ ❝♦♠ n ✈ért✐❝❡s✳ P♦r ✜♠✱ s❡ ♦ ❣r❛❢♦ t❡♠ ✉♠ ✈ért✐❝❡ ❛ ♠❛✐s✱ ♦✉ s❡❥❛✱ n+ 1 ✈ért✐❝❡s✱ t❡r❡♠♦s n ❛r❡st❛s ❛❞✐❝✐♦♥❛✐s✳ ❊♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s s❡rá n(n−1)

2 +n =

n2

−n+2n

2 =

n2

+n

2 =

(n+1)((n+1)−1) 2 ✳

❈❤❛♠❛r❡♠♦s ❞❡ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V(G) ♣✉❞❡r s❡r ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s V1(G) ❡ V2(G) ❞❡ ♠♦❞♦ q✉❡ t♦❞❛ ❛r❡st❛

❞♦ ❣r❛❢♦ t❡♠ ✉♠ ❡①tr❡♠✐❞❛❞❡ ❡♠ V1(G)❡ ❛ ♦✉tr❛ ❡♠ V2(G)✳

❯♠ ❣r❛❢♦ s✐♠♣❧❡s ❡ ❝♦♥❡①♦ ❝♦♠ n ✈ért✐❝❡s ❡ n − 1 ❛r❡st❛s s❡rá ❞❡♥♦♠✐♥❛❞♦ ❞❡ ár✈♦r❡✳ ❈❧❛r❛♠❡♥t❡ t❡♠♦s q✉❡✱ ❡♠ ✉♠❛ ár✈♦r❡✱ ♥ã♦ ❡①✐st❡♠ ❧❛ç♦s✱ ❛r❡st❛s ♠ú❧t✐♣❧❛s ♥❡♠ ✏❝✐❝❧♦s✑✳ ❯♠❛ ✢♦r❡st❛ é ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ❡ ❞❡s❝♦♥❡①♦✱ s❡♥❞♦ q✉❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ é ✉♠❛ ár✈♦r❡✳ ❯♠❛ ár✈♦r❡ ❣❡r❛❞♦r❛ ❞♦ ❣r❛❢♦ ❝♦♥❡①♦ G é ✉♠❛ ár✈♦r❡ T = (V(T), E(T))♦♥❞❡V(T) = V(G) ❡E(T)⊂E(G)✱ ♦✉ s❡❥❛✱ é ✉♠❛ ár✈♦r❡ ❝♦♠ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s q✉❡ G ❡ ❝♦♠ ❛r❡st❛s ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛s ❛r❡st❛s ❞❡G✳

v1

v2

v3

v4

v5 v6

✭❛✮ ●r❛❢♦ ❈♦♠♣❧❡t♦K5

v1

v2

v3

v4

v5

v6

v7

✭❜✮ ●r❛❢♦ ❇✐♣❛rt✐❞♦

v1

v2

v3

v4 v5

v6

✭❝✮ ●r❛❢♦ G

v1

v4 v5

v6

✭❞✮ ❙✉❜❣r❛❢♦ ❞❡G✭ár✈♦r❡✮

●r❛❢♦s ❡ ▼❛tr✐③❡s ✷✹

❯♠ ♣❡r❝✉rs♦vα1vα2. . . vα_k ❡♠ G é ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛❧q✉❡r ❞❡ ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s✱

♦♥❞❡ vα1 ❡ vαk sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ♦ ♣♦♥t♦ ✜♥❛❧ ❞♦ ♣❡r❝✉rs♦✳ ❯♠

♣❡r❝✉rs♦ é ❝❤❛♠❛❞♦ ❞❡ ❝❛♠✐♥❤♦ q✉❛♥❞♦ ♥ã♦ ❤á r❡♣❡t✐çã♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ✈ért✐❝❡s✳ ◗✉❛♥❞♦✱ ❡♠ ✉♠ ❝❛♠✐♥❤♦✱ t❡♠♦s vα1 = vα_k✱ ❞❡♥♦♠✐♥❛♠♦s ❝✐❝❧♦✳ ❯♠ ❧❛ç♦ é ✉♠ ❝✐❝❧♦

q✉❡ ❝♦♥té♠ ❛♣❡♥❛s ✉♠ ✈ért✐❝❡ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s♦♠❡♥t❡ ✉♠❛ ❛r❡st❛✳

❉❡✜♥✐çã♦ ✸✳✺✳ ❙❡❥❛♠ ❞♦✐s ❣r❛❢♦s G= (V(G), E(G)) ❡ H = (V(H), E(H))✳ ❉✐③❡♠♦s q✉❡ G ❡ H sã♦ ✐s♦♠♦r❢♦s s❡ ❡①✐st✐r ✉♠❛ ❜✐❥❡çã♦ ϕ : V(G) → V(H) t❛❧ q✉❡ ab ∈

E(G)⇔ϕ(a)ϕ(b)∈E(H)✳

●r❛❢♦s ❖r✐❡♥t❛❞♦s✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❣r❛❢♦ ♦r✐❡♥t❛❞♦ t❡♠ ❝♦♠♦ ✐♥t✉✐t♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ♦r✐❡♥t❛çã♦ ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s ❛r❡st❛s ❞❡ ✉♠ ❣r❛❢♦ q✉❛❧q✉❡r✳ ❉❡ ♠♦❞♦ ❢♦r♠❛❧ t❡♠♦s✿

❉❡✜♥✐çã♦ ✸✳✻✳ ❯♠ ❣r❛❢♦ ♦r✐❡♥t❛❞♦ ✭❞✐❣r❛❢♦✮ D é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♣❛r (V(D), E(D)) ♦♥❞❡ V(D) é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ♥ã♦✲✈❛③✐♦ ❞❡ ❡❧❡♠❡♥t♦s ❡ E(D) ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞✐st✐♥t♦s✳

◆❡st❡ ❝❛s♦✱ é ❝♦♠✉♠ ❞❡♥♦♠✐♥❛r ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦E(D)❞❡ ❛r❝♦s✱ ❡ q✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r ♥❡♥❤✉♠ r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦ t❛♠❜é♠ ♣♦❞❡♠♦s ❞❡♥♦♠✐♥á✲❧♦s ❞❡ ❛r❡st❛s✳ ◆♦t❡ q✉❡ ♠✉✐t❛s ❞❛s ♥♦t❛çõ❡s ❡st❛❜❡❧❡❝✐❞❛s ♣❛r❛ ❣r❛❢♦s t❛♠❜é♠ s❡rã♦ ✉t✐❧✐③❛❞❛s ❡♠ ❣r❛❢♦s ♦r✐❡♥t❛❞♦s✱ ♦❜✈✐❛♠❡♥t❡ ❧❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛❧❣✉♠❛s ♣❡❝✉❧✐❛r✐❞❛❞❡s✱ ❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ r❡❢❡r❡♥t❡ à ♦r✐❡♥t❛çã♦ ❞❛ ❛r❡st❛✳

P❛r❛ ✉♠ ❛r❝♦ (u, v) ♦ ♣r✐♠❡✐r♦ ✈ért✐❝❡ u é ❝❤❛♠❛❞♦ ❞❡ ❝❛✉❞❛ ♦✉ ♣♦♥t❛ ✐♥✐❝✐❛❧ ❡ ♦ s❡❣✉♥❞♦ ✈ért✐❝❡ v é ❝❤❛♠❛❞♦ ❞❡ ❝❛❜❡ç❛ ♦✉ ♣♦♥t❛ ✜♥❛❧✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❛r❝♦ (u, v) ❞❡✐①❛ u ❡ ❝❤❡❣❛ ❡♠ v✳ ❆ ❝❛❜❡ç❛ ❡ ❛ ❝❛✉❞❛ ❞❡ ✉♠ ❛r❝♦ sã♦ ❛❞❥❛❝❡♥t❡s ✭♦✉ ✈✐③✐♥❤♦s✮ ❡ ✐♥❝✐❞❡♠ ♥❡ss❡ ❛r❝♦✳ P♦❞❡♠♦s ❞❡♥♦t❛r ♦ ❛r❝♦ e = (x, y) ♣♦r xy ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦r e✳

❖ ❣r❛✉ ❞❡ ✉♠ ✈ért✐❝❡ ❡♠ ✉♠ ❣r❛❢♦ ♦r✐❡♥t❛❞♦ é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛r❡st❛s ✐♥❝✐❞❡♥t❡s ♥❡ss❡ ✈ért✐❝❡✳ ◗✉❛♥❞♦ ❢♦r ♥❡❝❡ssár✐♦ ✉s❛r❡♠♦s ❣r❛✉ ❞❡ ❡♥tr❛❞❛ ✭❣r❛✉ ❞❡ s❛í❞❛✮ ❞♦ ✈ért✐❝❡ v ♣❛r❛ ♥♦s r❡❢❡r✐r ❛♦s ❛r❝♦s ❝♦♠ ♣♦♥t❛ ✜♥❛❧ ✭♣♦♥t❛ ✐♥✐❝✐❛❧✮ ❡♠ v✳

✸✳✶ ●r❛❢♦s ❡ ▼❛tr✐③❡s

▼❛tr✐③ ❞❡ ❆❞❥❛❝ê♥❝✐❛ ❡ ❞❡ ■♥❝✐❞ê♥❝✐❛

●r❛❢♦s ❡ ▼❛tr✐③❡s ✷✺

❈❡rt❛♠❡♥t❡ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡ss❡s ❡st✉❞♦s ❡stã♦ ♥♦ ❝❛♠♣♦ ❞❛ ❝♦♠♣✉t❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❣r❛❢♦ ♣❛r❛ ✉♠ ❝♦♠♣✉t❛❞♦r é ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠❛ ♠❛tr✐③✳

❆ s❡❣✉✐r ❞❡✜♥✐r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❣r❛❢♦ ❡ ❛❞✐❛♥t❡ ❡s✲ t✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ t❡♥t❛♥❞♦ ❢❛③❡r ✉♠❛ ❝♦♥❡①ã♦ ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❛♣r❡s❡♥t❛❞♦s ❛té ❛❣♦r❛✳

❉❡✜♥✐çã♦ ✸✳✼✳ ❙❡❥❛ G ✉♠ ❣r❛❢♦ ❝♦♠ ✈ért✐❝❡s r♦t✉❧❛❞♦s v1, v2, . . . , vn✳ ❆ ♠❛tr✐③ ❞❡

❛❞❥❛❝ê♥❝✐❛ A(G) ❞❡ G é ❛ ♠❛tr✐③ n ×n ♥❛ q✉❛❧ ❝❛❞❛ ❡♥tr❛❞❛ aij ❝♦rr❡s♣♦♥❞❡ ❛♦

♥ú♠❡r♦ ❞❡ ❛r❡st❛s ✐♥❝✐❞✐♥❞♦ ❡♠ vi ❡ vj✳

❆ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❞❡ ✉♠ ❣r❛❢♦ é s✐♠étr✐❝❛ ❝♦♠ r❡❧❛çã♦ à ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✳ ❚❛♠❜é♠✱ ♣❛r❛ ✉♠ ❣r❛❢♦ s❡♠ ❧❛ç♦s✱ ❝❛❞❛ ❡♥tr❛❞❛ ❞❛ ❞✐❛❣♦♥❛❧ é ✐❣✉❛❧ ❛ ✵✳ ❆ s♦♠❛ ❞❛s ❡♥tr❛❞❛s ❡♠ q✉❛❧q✉❡r ❧✐♥❤❛ ✭❝♦❧✉♥❛✮ é ♦ ❣r❛✉ ❞♦ ✈ért✐❝❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥❛q✉❡❧❛ ❧✐♥❤❛ ✭❝♦❧✉♥❛✮✳

❆s ♠❛tr✐③❡s ❞❡ ❛❞❥❛❝ê♥❝✐❛ A(G) ❡ A(H) ❛❜❛✐①♦ r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ❣r❛❢♦s G❡ H ❞❛s ✜❣✉r❛s ✸✳✷❛ ❡ ✸✳✷❜✳ P❛r❛ ✉♠❛ ♠❡❧❤♦r ❝❧❛r❡③❛✱ ✐♥❞❡①❛♠♦s ❛s ❧✐♥❤❛s ❡ ❛s ❝♦❧✉♥❛s ❞❛s ♠❛tr✐③❡s ❛♦s ✈ért✐❝❡s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❉❡✐①❛♠♦s ❝❧❛r♦ q✉❡ ❛ ♠❛tr✐③ é ❝♦♠♣♦st❛ ❛♣❡♥❛s ♣❡❧♦s ♥ú♠❡r♦s ✏❞❡♥tr♦✑ ❞♦s ♣❛rê♥t❡s❡s✳ ❋❛r❡♠♦s ♦ ✉s♦ ❞❡ss❡ r❡❝✉rs♦ ❛♦ ❧♦♥❣♦ ❞❛s ♣á❣✐♥❛s s❡❣✉✐♥t❡s✳

A(G) =        

v1 v2 v3 v4 v5

v1 0 1 0 1 1

v2 1 0 1 1 0

v3 0 1 0 1 0

v4 1 1 1 0 0

v5 1 0 0 0 0        

A(H) =          

a1 a2 a3 a4 a5 a6

a1 0 1 1 1 0 1

a2 1 0 1 1 0 0

a3 1 1 0 1 1 0

a4 1 1 1 0 1 1

a5 0 0 1 1 0 0

a6 1 0 0 1 0 0          

v5 v1

v4

v2

v3

✭❛✮ ●r❛❢♦G

a1

a2

a3

a4 _a

5

a6

✭❜✮ ●r❛❢♦H

●r❛❢♦s ❡ ▼❛tr✐③❡s ✷✻

P❛r❛ ❣r❛❢♦s ❞❡s❝♦♥❡①♦s✱ ✈❛❧❡ ♥♦t❛r q✉❡ ❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ é ✉♠❛ ♠❛tr✐③ ❡♠ ❝♦♠♣♦st❛ ♣♦r ❜❧♦❝♦s✱ ♦♥❞❡ ❝❛❞❛ ❜❧♦❝♦ é ✉♠❛ ♠❛tr✐③ ❛❞❥❛❝ê♥❝✐❛ ❞❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡✳ P♦r ❡①❡♠♣❧♦✱ A=                  

b1 b2 b3 b4 b5 b6 b7 b8 b9

b1 0 1 1 0 1 0 0 0 0

b2 1 0 1 0 0 0 0 0 0

b3 1 1 0 1 0 0 0 0 0

b4 0 0 1 0 0 0 0 0 0

b5 1 0 0 0 0 0 0 0 0

b6 0 0 0 0 0 0 1 1 1

b7 0 0 0 0 0 1 0 1 1

b8 0 0 0 0 0 1 1 0 0

b9 0 0 0 0 0 1 1 0 0

                 

❝♦rr❡s♣♦♥❞❡ ❛♦ ❣r❛❢♦

b1 b2 b3 b₄ b₅ b₆ b7

b8 b9

▲❡♠❛ ✸✳✽✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ s✐♠♣❧❡s ❝♦♥❡①♦ G✱ ❝♦♠ n ✈ért✐❝❡s ❡ s✉❛ ♠❛tr✐③ ❛❞❥❛❝ê♥❝✐❛

A✱ ❡♥tã♦ ❛ ❝♦♠♣♦♥❡♥t❡ cij ❞❡ Ak✱ ♣❛r❛ ❛❧❣✉♠ k ✐♥t❡✐r♦ ❡ ♣♦s✐t✐✈♦✱ é ♦ ♥ú♠❡r♦ ❞❡

❝❛♠✐♥❤♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ k q✉❡ ❧✐❣❛ ♦ ✈ért✐❝❡ vi ❛♦ ✈ért✐❝❡ vj✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ k = 1 ❛ ❛✜r♠❛çã♦ é ó❜✈✐❛✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ♠❛tr✐③ ❛❞❥❛❝ê♥❝✐❛ ❡①✐st❡aij =aji = 1s❡vi❡vj sã♦ ❛❞❥❛❝❡♥t❡s✳ P❛r❛k= 2♦ ❡❧❡♠❡♥t♦cij ❞❡A2 s❡r❛ ❞❛❞♦✱

♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ♣♦rcij = n

P

l=1

ailalj ♥♦t❡ q✉❡ ❛s ♣❛r❝❡❧❛s s❡rã♦

♥ã♦ ♥✉❧❛s s♦♠❡♥t❡ q✉❛♥❞♦ ✉♠ ✈ért✐❝❡ vl✱ l = 1, . . . , n✱ ❢♦r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛❞❥❛❝❡♥t❡

❛♦ ✈ért✐❝❡ vi ❡ ♦ ✈ért✐❝❡ vj✳ ❚❛♠❜é♠ ✈❛❧❡ ♥♦t❛r q✉❡ ♦ ❡❧❡♠❡♥t♦ cii ❞❡ A2 ✈ért✐❝❡ é ♦

❣r❛✉ ❞♦ ✈ért✐❝❡ vi✳ P❛r❛ k= 3 ❜❛st❛ ♦❜s❡r✈❛r q✉❡A3 =A2·A✱ ✐♠♣❧✐❝✐t❛♠❡♥t❡ t❡♠♦s

♦s ❝❛♠✐♥❤♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✷ ♠❛✐s ♦s ❝❛♠✐♥❤♦s ✭q✉❡ sã♦ ❛❞❥❛❝❡♥t❡s✮ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✶✳ P♦❞❡♠♦s ❡♥tã♦ ❡st❡♥❞❡r ❡ss❛ ❧✐♥❤❛ ❞❡ r❛❝✐♦❝í♥✐♦ ♣❛r❛ Ak✳

❉❡✜♥✐çã♦ ✸✳✾✳ ❙❡❥❛ G ✉♠ ❣r❛❢♦ s✐♠♣❧❡s✱ ❝♦♠ n ✈ért✐❝❡s r♦t✉❧❛❞♦s v1, v2, . . . , vn ❡ m

❛r❡st❛s r♦t✉❧❛❞❛s e1, e2, . . . , em✳ ❆ ♠❛tr✐③ ❞❡ ✐♥❝✐❞ê♥❝✐❛Q(G) é ❛ ♠❛tr✐③n×m ♥❛ q✉❛❧

❝❛❞❛ ❡♥tr❛❞❛ é✿

qij =

 



1 s❡ ♦ ✈ért✐❝❡ vi é ✐♥❝✐❞❡♥t❡ ❛ ❛r❡st❛ ej✱

0 ❝❛s♦ ❝♦♥trár✐♦✳