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The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society.

Volume 7, pp. 152-173, September 2000.

ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

✂✁☎✄✝✆✟✞✡✠✡☛✌☞✡✍✏✎✌✑✒✡✑✓✎✕✔✖✞✂✗✓✘✙✁✚✍✏✛✢✜✣✔✤✠✥✂✑✦✄✧✎✌✠★✁✚✗✓✆✩✔✖✠✪✄✬✫✭✄✮✆✯✑✱✰✲✄ ☛✯✫✳✰✴✑✵☞✂✄✝✞✡✠✡✑✱✰✶✑✱✍✭✆✷✜✸✑✱✑✵✡☛✩☞✤✎✌✹✏✺

✻✧✼✾✽❀✿❂❁❄❃✬❅✧❆✧❇✩❈❊❉❊❋❍●✧■✧✽❑❏▲✼❊▼❖◆✦✼❊❅P▼P◆✦◗✧❁✧❁✧❘

❙❯❚❑❱❳❲❩❨❩❬❪❭❫❲❵❴ ✿❜❛✓❝❡❞❣❢✐❤❦❥❣❧❫❥♥♠❳♦q♣r❝s❞❣♠✉t❣❢✐❤✈❝❡✇①♦❡②③❧❫❛❣④❳♠✉t①♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✵❥❣♦s⑤⑥②❣⑦✐♠❳⑨❍⑩❶⑤⑥♦✧t❣♠❳❤❡④❳♦❡❢✐❥①❝s⑤⑥♦❦❤s❷☎❤s❝s♠❳⑨✸❤✧❸☎❢❹❧❯⑩❺♠❳♠qt☎❻ ②③❧❫④❽❼❵❤P❝s❞③❧❩❝▲❧❫♦❡♠❦②③❧❫❤s♠qt✓⑤⑥❛✱⑨✸♠q❧❫❤❡✇❣♦s♠q⑨✸♠❳❛✚❝❡❤✝❢✐❤✝t❣❢✐❤❡④❳✇❣❤s❤❡♠❳t❿❾✬❁✝♠❳④q♠❳❤s❤❽❧❫♦s❷✵❧❫❛❣t✦❤s✇①➀✸④❳❢✐♠❳❛❵❝✧④❳⑤⑥❛❣t❣❢❝s❢✐⑤⑥❛❣❤▲❧❫♦s♠❦t❣♠❳♦❡❢❸✚♠❳t ⑩❺⑤⑥♦➁❝s❞❣♠✵♠➃➂☎❢✐❤s❝s♠q❛①④q♠✱⑤❫⑩P⑩❺♠❳♠❳t❣②③❧❫④❽❼❵❤✸❤❡⑤✓❝s❞③❧❩❝✉❝s❞❣♠✵t❣❢✐❤✈❝❡✇①♦❡②③❧❫❛❣④❳♠❳❤✉t❣⑤✟❛❣⑤❫❝✉❢✐❛①➄③✇❣♠❳❛❣④❳♠✱❝s❞❣♠✱❢✐❛❣❥❣✇①❝❽➅❩⑤⑥✇①❝s❥❣✇①❝❯②❪♠❳❞③❧q❸❵❻ ❢✐⑤⑥♦➆❾✝◗✬❝✧❝s❞❣♠✙❤❽❧❫⑨✸♠❦❝s❢✐⑨✸♠✚♣①❝❡❞❣♠➁④❳⑦✐⑤⑥❤❡♠❳t①❻❜⑦✐⑤❵⑤⑥❥✓❤s❷☎❤s❝s♠❳⑨➇②♥♠❳④❳⑤⑥⑨✸♠❳❤✧♦s♠q⑧⑥✇①⑦❹❧❫♦➆♣♥⑤❫⑩➈❢✐❛①t❣♠➃➂✓❧❩❝▲⑨✸⑤⑥❤✈❝✧⑤⑥❛①♠✚♣r❧❫❛❣t✓❥❪⑤⑥❤s❤❡❢✐②❣⑦❷ ❧❫⑦✐❤❡⑤✯❤s❝❡❧❫②❣⑦✐♠⑥❾★➉✮❞❣♠✵④❳⑤⑥❛❣❤✈❝❡♦s✇❣④➃❝❡❢✐⑤⑥❛✡❢✐❤✉②③❧❫❤❡♠❳t★⑤⑥❛★❧✟❝s♦❽❧❫❛❣❤✈⑩❶⑤⑥♦s⑨❯❧❩❝❡❢✐⑤⑥❛✕❝s⑤✩④❳⑤⑥❛❣t①♠q❛①❤❡♠❳t★⑩❺⑤⑥♦❡⑨❄✇①❛❣t❣♠❳♦✸⑤⑥♦s❝s❞❣⑤⑥⑧⑥⑤⑥❛③❧❫⑦ ♠❳➊☎✇①❢❸⑥❧❫⑦✐♠❳❛❣④❳♠❦❝❡♦❡❧❫❛❣❤s⑩❺⑤⑥♦❡⑨❯❧❩❝s❢✐⑤⑥❛❣❤✬❝❡❞❣❧❩❝✧④q❧❫❛✵②❪♠❦t❣❢✐♦s♠❳④❳❝s⑦❷➋❢✐⑨✸❥❣⑦✐♠❳⑨✸♠❳❛❵❝s♠❳t✵❧❫❤✝❧✣❛❵✇❣⑨✸♠❳♦❡❢✐④q❧❫⑦✐⑦❷❯❤s❝❡❧❫②❣⑦✐♠❦⑨✸♠➃❝s❞❣⑤☎tr❾

➌✵➍❵➎➐➏➁➑ ❨❫➒❑❱⑥❴

✻✧❢✐❤s❝s✇❣♦❡②❣❧❫❛❣④❳♠✦t❣♠❳④❳⑤⑥✇❣❥❣⑦✐❢✐❛❣⑧❣♣✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✸❤s❷☎❤s❝s♠q⑨✸❤q♣✝■✧♦s❝s❞❣⑤⑥⑧⑥⑤⑥❛③❧❫⑦✝❝❡♦❡❧❫❛❣❤s⑩❺⑤⑥♦❡⑨❯❧❩❝s❢✐⑤⑥❛❿♣✬❃✬⑤⑥❛☎❻ t❣♠❳❛❣❤s♠qt✱⑩❶⑤⑥♦s⑨

❙❯➓→➔★❱❩➣❀❚❿↔➍ ❭❫❲❯❭❵↕➙❬❪❱q❱❩➛✐➜➝❭⑥❬❣❲⑥➛➑❿➞ ❱⑥❴✧➟⑥➠⑥➡✬➢⑥➤ ♣ ➟⑥➠⑥➡❊➥✚➢ ♣ ➟⑥➠⑥➡✬➤⑥➦ ♣❪➧ ➤❫➨➝➠⑥➤

➪❰➹✓➴❮Ô✕➷☎➴r➹q✃➶➬❩➷❵Ô✕➷☎➽r➾①➮ ïûô✲õ✉ü

➪❰➹✕➾➆➚➝➷✏Ñ♥✃➶➾➆➘➶✃❑➾①➮❯➴❪➽➝❐

ëýôþõ❯ÿ

➪❰➹✤➴ù❐❑➪❰➹❳➾❩✃➶➬❩Ï➝➴❪➽➝Ð❵➷❮➾➆➚➝➴❪➾★Ô✖➴❣❒è➬❩➷❵➘➶➬❩➷☎➹➆➷❵➽❿➾✤Ô✕Ñ❑❐❑➷❵Ò❶➪❶➽➶Ó Ñ♥➬✸Ô✕➷①➴♥➹➆✃➶➬➆➪❶➽➶Ó✕➷❵➬❩➬❩Ñ♥➬❫➹❵➮❀➽➶Ñr➪❶➹➆➷♥➮➶➚➝➪❺Ór➚➶➷❵➬✁❡Ñr➬❩❐➶➷❵➬➁➾➆➷❵➬❩Ô✖➹✸➪❶➽❮➴✕Ò❶➪❶➽➶➷☎➴♥➬➆➪

(2)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò❁❀✾✥

Ñ♥➬✬➾❩➚➶➷✙Ð❵Ò❺Ñ❿➹q➷①❐✖❡Ò❶Ñ❀Ñ♥➘✦➹q❒❑➹q➾➆➷☎Ô✂Ý✝PP➪❶➬❫➹❳➾✝Ñ

Õ

Ý❩✆✉➚➶➪❰➹➁Ð❵Ñ♥➽➝❐❑➪❺➾➆➪❶Ñ♥➽✖Ór✃➝➴❪➬❫➴❪➽❿➾❩➷❵➷☎➹❦➾➆➚➝➴❪➾✙➾➆➚➝➷✱Ð✚Ò❶Ñr➹➆➷☎❐◗❡Ò❶Ñ❀Ñ♥➘✖➹q❒❑➹q➾➆➷☎Ô✲➚➝➴r➹✙➴✯✃➶➽➶➪❬❈r✃➝➷➋➹qÑrÒ❺✃➶➾➆➪❶Ñ♥➽

Õ

Ñ♥➬▲➴❪Ò❶Òr➹➆✃◗❭✤Ð✚➪❶➷❵➽❿➾❩Ò❺❒✟➹➆Ô✕Ñ❿Ñ♥➾➆➚

ë

➴❪➽✾❐✟➴❪➘➶➘➝➬➆Ñr➘➶➬➆➪❰➴③➾❩➷❦➪❺➽➝➪➙➾❩➪❶➴♥Ò❀Ð❵Ñ♥➽➝❐➶➪➙➾❩➪❺Ñr➽➝➹❵Ý✿❪❀➷☎Ð❵Ñ♥➽➝❐❑Ò❶❒♥➮③➱✣➷✙➱✣Ñ♥✃➝Ò❶❐✦Ò❶➪❺Ör➷ ➾➆➚➝➷✉Ð❵Ò❺Ñ❿➹q➷①❐✖❽Ò❺Ñ❀Ñ♥➘✌➹q❒❑➹q➾➆➷❵Ôý➾❩Ñ✦➚➝➴❣×♥➷❴❫❛❵▲❜❝❙❡❞❢❲❨❣✝❤❢✐❨❥❦❣☎✐❨❵❧❙✚➮❪➪❡Ý➷♥Ý❶➮♥➴❪Ò❶Ò❿➾➆➚➶➷♥♠rÑ♥➬❫❐➶➴❪➽✟Ï➝Ò❺Ñ❑Ð❫Ö❑➹▲➴♥➹❩➹qÑ❑Ð✚➪❰➴③➾❩➷☎❐

(3)

ELA

✃➶➬➁➽➶➷☎Ð❵➷☎➹❩➹➆➴♥➬➆❒✕➴❪➽➝❐★➹q✃◗❭✤Ð❵➪❺➷☎➽r➾✉Ð✚Ñr➽➝❐❑➪❺➾➆➪❶Ñ♥➽➝➹➁➱✸➪❺Ò❶Ò❊❐❑➷❵➘➈➷❵➽➝❐★Ñ♥➽✖➾❩➚➶➷❴❈❿✃➝➴♥➽❿➾➆➪❺➾➆➪❶➷☎➹✙➾➆➚➝➴❪➾✉Ð☎➴❪➽✤Ï➈➷ ➬❩➷☎➴♥❐✡Ñ✾❮ ➴♥➹✕➱✣➷❵Ò❶Ò❯➴♥➹✩Ór➷❵Ñ♥Ô✕➷❵➾➆➬❩➪❶Ð✂Ð✚Ñr➽➝❐❑➪❺➾➆➪❶Ñ♥➽➝➹✕➬❩➷❵Ò❰➴③➾❩➷☎❐➐➾❩Ñù❐❑➪❶Ô✩➷☎➽➝➹➆➪❺Ñr➽➝➹✩Ñ

Õ

➹q✃➶Ï✾➹q➘➝➴rÐ✚➷①➹❵ÝÏ✆✉➚➶➪❶➹★➴❪Ò❶Ò❺Ñ③➱❯➹✩➴ Ð✚Ñ❀Ñr➬❩❐❑➪❶➽➝➴❪➾➆➷✪

P✧➪❺➽✾➴❪Ò❶Ò❺❒r➮♥➱✣➷✸➚➝➴❣×♥➷✉➾❩➚➶➷➋Ð❫➚➝➴♥➬❩➴rÐ⑥➾➆➷☎➬➆➪

✂ ➪❶➽➝❐❑➷✪♣❮➴③➾✸Ô✕Ñr➹q➾❯Ñ♥➽➶➷✓➴♥➽➝❐✡➴♥Ò❶➹➆Ñ✕➱✸➚➶➷❵➽✂➪❺➾✱Ð❵➴♥➽✡Ï➈➷✓Ô✖➴♥❐❑➷✓➹q➾❩➴❪Ï➝Ò❺➷✦Ï❀❒

(4)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò❁❀❝❀ ➽❀✃➶Ô✕➷❵➬❩➪❶Ð☎➴❪Ò❶Ò❺❒✟➹q➾❩➴♥Ï➶Ò❶➷➁Ô✕➷✚➾❩➚➶Ñ❑❐➶➹❦➴♥➹▲➹➆✃➶Ó♥Ór➷☎➹q➾➆➷①❐✓➪❺➽❮Ø

(5)

ELA

Ð✚Ñr➽➝➹q➾➆➬❩✃➝Ð⑥➾❩➪❺×r➷❵Ò❶❒✖Ï❿❒✤➾➆➚➶➷

(6)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

(7)
(8)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

(9)

ELA

➪❶➽✂Ñr✃➶➬✸Ô✖➴❪➪❶➽★➾❩➚➶➷❵Ñr➬➆➷☎Ô✖➹✉➪❺➽✂➾➆➚➝➷✓➽➶➷✪♣❀➾➋➹➆➷☎Ð⑥➾❩➪❺Ñr➽✮Ý

❋✝➫

✰❍●➶➻✈➭➇➲❨æ✖ç①➸❅■✈➯❉ç

➫➋➼➃➽❮➾➆➚➝➪❶➹✱➹q➷①Ð⑥➾➆➪❶Ñ♥➽✝➮➝➱✣➷✓➘➶➬➆➷①➹q➷☎➽❿➾❯Ñ♥✃➶➬✸➾❳➱✣Ñ✖Ô✖➴❪➪❶➽❮➾➆➚➶➷☎Ñ♥➬❩➷❵Ô✖➹☎Ýr✆✉➚➶➷➨♦✾➬❩➹q➾ ➾➆➚➝➷❵Ñ♥➬❩➷❵ÔþÓ♥➪❶×♥➷①➹ ➽➶➷①Ð✚➷①➹➆➹❩➴❪➬❩❒✌➴♥➽➝❐✕➾➆➚➝➷✱➹➆➷☎Ð✚Ñr➽➝❐✖➹➆✃◗❭✤Ð✚➪❶➷❵➽❿➾✣Ð❵Ñ♥➽➝❐❑➪❺➾➆➪❶Ñ♥➽✾➹

Õ

Ñr➬ ➾➆➚➝➷➋➷✪♣❑➪❰➹❳➾❩➷❵➽➝Ð❵➷➋Ñ Õ✬Õ

➷❵➷☎❐◗ Ï➝➴rÐ❫Ö❀➹✮➾➆➚➝➴❪➾✧Ô✖➴♥Ö♥➷➁➴✱➹q❒❑➹q➾➆➷☎Ôû➬❩➷❵Ór✃➶Ò❰➴❪➬①➮☎Ñ

Õ

➪❺➽✾❐❑➷✪♣✟➴③➾✧Ô✕Ñr➹q➾✧Ñr➽➶➷♥➮③➴❪➽✾❐✯➹q➾❩➴❪Ï➝Ò❺➷➁✃➝➹q➪❶➽➶Ó✱Ô✩➷①➴♥➹➆✃➶➬❩➷❵Ô✕➷❵➽❿➾

(10)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

(11)
(12)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò

Ù

➱✸➪❺➾➆➚ ❳

ì

✮✒Ñr➬q➾❩➚➶Ñ♥ÓrÑ♥➽➝➴♥Òs➮

(13)
(14)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò

➪❶➹✵➽➶Ñr➽➝➹q➪❶➽➶Ór✃➶Ò❶➴♥➬☎➮

(15)
(16)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

(17)
(18)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò

✃➶Ò❺ÒPÐ✚Ñ♥Ò❶✃➶Ô✕➽❮➬❩➴♥➽➶Ö❊➮❑➬➆➷①➹q➘➈➷☎Ð✚➾➆➪❶×♥➷☎Ò❺❒r➮❀➹➆Ñ✩➾➆➚➶➷☎➬➆➷✓➷✹♣❑➪❶➹q➾❩➹✸➴❪➾❯Ò❺➷①➴♥➹q➾✉Ñr➽➶➷

(19)

ELA

➦sr✎t

(20)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò

➦sr✎t

(21)

ELA

➪❶➽➝❐❑➷✹♣✂➴❪➾➋Ô✕Ñr➹q➾❯Ñ♥➽➶➷rÝ✱②➇➪➙➾❩➚

à Ð✚Ñr➽➝❐❑➪❺➾➆➪❶Ñ♥➽➝➹✉➪❶➽✂➾➆➚➶➷①➹q➷✦➾❩➚➶➷❵Ñr➬➆➷☎Ô✖➹➁➘➝➬➆➷①➹q➷☎➽r➾❯Ð❵Ñ♥Ô✕➘➶Ò❶➷✚➾➆➷✓➽➝➷☎Ð✚➷①➹➆➹❩➴❪➬❩❒✤➴❪➽✾❐✡➹➆✃◗❭✤Ð✚➪❶➷❵➽❿➾➋Ð✚Ñr➽➝❐❑➪❺➾➆➪❶Ñ♥➽➝➹☎Ý

➼➃➽✡Ð❵Ñ♥➽❿➾➆➬❫➴♥➹q➾ ➾❩Ñ✟➾➆➚➝➷✵Ð❵Ñ♥➽➝❐❑➪❺➾➆➪❶Ñ♥➽✾➹✙➪❶➽Ó⑨✝➷❵Ô✕Ô✖➴♥➹ Ð✚Ñr➽➝❐❑➪❺➾➆➪❶Ñ♥➽➝➹✸➴♥➬➆➷✦➪❶➽❿➾➆➷❵➬➆➾❳➱✸➪❶➽➶➷☎❐★➾➆Ñ✖Ór✃➝➴❪➬❫➴❪➽❿➾➆➷☎➷✱➹➆➷❵×r➷❵➬❫➴❪Ò❊➘➶➬❩Ñ♥➘➈➷❵➬➆➾➆➪❶➷☎➹✸➴③➾✸Ñr➽➝Ð✚➷rÝ

①P➫ ✎✌➳➈➭P➺❆■✈➸✿ç☎➻s➳✾➭✿ç ➫❷②→➷✯➚➝➴❣×r➷✓➘➶➬➆➷①➹q➷☎➽❿➾➆➷☎❐✂➽➶➷①Ð✚➷①➹➆➹❩➴❪➬❩❒★➴❪➽✾❐✭➹q✃◗❭✤Ð❵➪❺➷☎➽r➾✵Ð✚Ñ♥➽✾❐❑➪➙➾❩➪❺Ñr➽➝➹ ➬❩➷☎➹➆✃➶Ò➙➾❩➪❺➽➝Ó✕Ð❵Ò❺Ñ❿➹q➷①❐✖❽Ò❺Ñ❀Ñ♥➘✂➹q❒❑➹q➾➆➷☎ÔßÏ✾➷✓➬❩➷❵Ór✃➶Ò❰➴❪➬①➮❿Ñ

Õ

★④③➋✼P❾➈◗✧❛❣t❣♠❳♦s❤❡⑤⑥❛❿♣❬⑤➶❾

❧❫❢❰♣✧❃✧❾

❢✐❤s④❽❞❣⑤❫⑩✈♣⑦⑥❣❾⑧✏❾✮✻✧♠❳⑨✸⑨✸♠❳⑦❰♣✪⑥❣❾✮✻✧⑤⑥❛❣⑧✚❧❫♦s♦❽❧①♣❅⑥❣❾✮✻➁❾✝❃❊♦s⑤✳✮✚♣✝◗✙❾â✼✧♦❡♠❳♠❳❛☎②❣❧❫✇❣⑨✦♣ ✰❪❾♥❅✧❧❫⑨✸⑨❯❧❫♦❡⑦✐❢✐❛①⑧❣♣①◗✙❾❪◆✦④q❏▲♠q❛①❛❣♠➃❷❵♣✠✰♥❾❪■✧❤s❝s♦❡⑤⑥✇❣④❽❞①⑤➆❸♥♣❿❧❫❛①t✟✻➁❾❝✰☎⑤⑥♦s♠q❛①❤❡♠❳❛❿❾⑦⑨✧⑩⑦❶❆⑩❁❷❱❸✻❹❱❺④❻✑❼④❺✝❽❱❾✂❿➁➀♠➂✎❻q❾ ✰❵⑤☎④❳❢✐♠➃❝❜❷✵⑩❺⑤⑥♦✬✿❜❛❣t❣✇❣❤✈❝❡♦s❢❹❧❫⑦❑❧❫❛❣t✱◗✧❥❣❥❣⑦✐❢✐♠❳t✵◆✓❧❩❝❡❞❣♠❳⑨❯❧❩❝s❢✐④❳❤➆♣✙➃✾❞❣❢✐⑦❹❧❫t❣♠❳⑦✐❥❣❞❣❢❹❧①♣✠★ ➟⑥➟⑥➦ ❾

✇❣❛❣❤❡♠➃❻✦✼✧♠❳♦s❤s❝s❛❣♠❳♦➆♣③●✣❾❣◆✦♠❳❞❣♦❡⑨❯❧❫❛❣❛r♣③❧❫❛❣t✓❁ ❾❏✣❾③❁✝❢✐④❽❞❣⑤⑥⑦✐❤➆❾❊▼✝♠❳⑧⑥✇❣⑦❹❧❫♦s❢ã✮q❧❩❝❡❢✐⑤⑥❛✦⑤❫⑩➶t①♠q❤s④❳♦❡❢✐❥①❝s⑤⑥♦✧❤✈❷①❤✈❝❡♠❳⑨✸❤✮②❵❷ ⑤⑥✇☎❝❡❥❣✇☎❝✧⑩❺♠q♠❳t❣②❣❧❫④❽❼♥❾✶➅➆➤✲➤✲➤❞➥➎❼➆➙➌➋➎❺✓➉✧⑩⑦❿✝➍✩➊➌➓✯➙➌➍✜➉✭❷↕➊➌➋➎➍✬❼➆➊➌➏♣❪◗P❃

➧P③✱✻➁❾❊❃✬❞❵✇✡❧❫❛❣t★●➁❾❊◆✦♠❳❞❣♦s⑨❯❧❫❛❣❛❿❾✂✻✧❢✐❤s❝s✇❣♦s②③❧❫❛❣④❳♠✓t❣♠❳④❳⑤⑥✇❣❥❣⑦✐♠❳t✂⑤⑥②❣❤s♠❳♦s❸✚♠❳♦❯t❣♠❳❤❡❢✐⑧⑥❛✤⑩❶⑤⑥♦✸t❣♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦➋❤✈❷①❤✈❝❡♠❳⑨✸❤q❾ ➄✢➦➌❺❙➍✩❻✑➓❀❺→❷◗➊➌➋➎➍✬❼➆➊➌➏➧⑨➧❻✑➍✬➍✩➉❹♣ ➠➌➣ ✫➠✪✩➟↔❵➥✖➣ ♣✖★ ➟⑥➟⑥➟ ❾

②✩ ③✱✻➁❾r❃❊❞☎✇✟❧❫❛①t✦●✣❾♥◆✦♠❳❞❣♦❡⑨❯❧❫❛①❛❿❾P✻✧❢✐❤s❝s✇❣♦❡②❣❧❫❛❣④❳♠➁t❣♠❳④❳⑤⑥✇❣❥❣⑦✐❢✐❛①⑧➋⑩❺⑤⑥♦✧t❣♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✧❤s❷☎❤s❝s♠❳⑨✸❤➆❾❬➄✙➅➆⑩❬➇➨➈✙➉⑦❷↕➊➌➋➎➍✬❼➆➊➌➏ ➐❱➑➒➍✬➀➔➓→➉✐♣

(22)

ELA

✻P❢✐❤s❝s✇❣♦❡②③❧❫❛①④q♠ ✻P♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✉⑩❶⑤⑥♦✝✻✧♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✿✰✚❷①❤✈❝❡♠❳⑨✸❤✮②❵❷✱◆✦♠q❧❫❤❡✇①♦❡♠❳⑨✸♠❳❛❵❝ ➨ ♠❳♠❳t❣②③❧❫④❽❼

ò

Ú

➟ ③➋✽➈❾❿✻▲❧❫❢❰❾✰➄✢➀➔➋✝➜✖❿✝➏➙➌❼➩❷↕➊✖➋➎➍❫❼➫➊➌➏◗➄✢➦➌❺❙➍✩❻✑➓❀❺❽♣❿❸✚⑤⑥⑦✐✇❣⑨✸♠❴★✳★ ➣ ⑤❫⑩⑦⑨➧❻➛➭✑➍❫❿➁❼➆❻→➯❴➊➌➍✩❻✑❺✯➀➔➋✕❷↕➊➌➋➎➍✬❼➆➊➌➏✂➙➌➋➒➂❢➅➛➋✑➲④➊✖❼④➓❁➙➌➍✬➀❫➊➌➋ ➄❆➭➟➀❫❻➟➋➒➭➛❻✑❺❡❾❧✰❵❥❣♦❡❢✐❛❣⑧⑥♠❳♦✈❻❰●➈♠❳♦s⑦❹❧❫⑧❣♣ ➡ ♠❳♦❡⑦✐❢✐❛❿♣❝★➟➌➣⑥➟ ❾

②★ ➢ ③❢⑥❣❾⑧ ❾r✻✧♠❳⑨✸⑨✸♠❳⑦❑❧❫❛❣t ➡ ❾r❏✰➳❧❫⑧⑥❤s❝s♦✁✺⑤⑥⑨✦❾✧➉✮❞❣♠✣⑧⑥♠❳❛❣♠❳♦❽❧❫⑦✐❢ã✮❳♠❳t❅✰☎④❽❞❵✇❣♦❦t❣♠❳④❳⑤⑥⑨✸❥❪⑤⑥❤❡❢❝s❢✐⑤⑥❛✓⑤❫⑩✬❧❫❛✯❧❫♦❡②❣❢❝s♦❽❧❫♦✈❷✦❥❪♠❳❛❣④❳❢✐⑦ ➵❁➸⑦➺✢➻

✫➶▼✝⑤⑥②①✇❣❤s❝➶❤s⑤❫⑩❺❝❛✭✮❧❫♦s♠✟✭✝❢❝s❞✙♠❳♦s♦❡⑤⑥♦❀②❪⑤⑥✇❣❛❣t①❤➶❧❫❛❣t✣❧❫❥①❥❣⑦✐❢✐④q❧❩❝❡❢✐⑤⑥❛❣❤q❾P➃➈❧❫♦s❝❑✿✻✫q❝s❞❣♠❳⑤⑥♦s❷✙❧❫❛❣t✙❧❫⑦✐⑧⑥⑤⑥♦❡❢❝s❞❣⑨✸❤q❾ ⑩❁❷✧➇➼➥➎❼➆➙✖➋➎❺④➉❱➇➽➙➌➍♠➾✙➉❱➄❆➊➫➲➛➍✬➚◗➙✖❼➆❻q♣✖★

✫ã★q➧ ➢✑↔

✩❩➥ ♣◗★

➟⑥➟⑥➠

★✳★④③❉➃➝❾☎●❊❧❫❛✵✻✧⑤❵⑤⑥♦❡♠❳❛❿❾➝➉✮❞❣♠❦⑧⑥♠❳❛❣♠❳♦❽❧❫⑦✐❢ã✮❳♠❳t✵♠❳❢✐⑧⑥♠❳❛❣❤s❝s♦❡✇❣④➃❝s✇❣♦❡♠ ❥❣♦s⑤⑥②❣⑦✐♠❳⑨â❢✐❛➋⑦✐❢✐❛❣♠q❧❫♦✮❤s❷☎❤s❝s♠q⑨è❝❡❞①♠q⑤⑥♦✈❷✚❾✂➅➆➤✲➤◗➤❩➥➎❼➫➙➌➋➎❺✓➉ ⑩⑦❿✝➍✩➊➌➓✯➙➌➍✜➉❅❷↕➊➌➋➎➍✬❼➆➊➌➏♣③◗P❃➈❻➦ ➧✇✫ã★✳★✳★ ↔ ★ ➦⑥➟ ♣◗★ ➟➌➣ ★⑥❾

★ ➦ ③✱✽➈❾▼❦❾ ➨ ⑦✐♠➃❝s④❽❞❣♠❳♦➋❧❫❛❣t✤◗➁❾➝◗✧❤❡❧❫♦❽❧⑥❧❫❢❰❾★■✧❛✤t❣❢✐❤✈❝❡✇①♦❡②③❧❫❛❣④❳♠✦t①♠q④❳⑤⑥✇❣❥①⑦✐❢✐❛❣⑧✌❢✐❛✤t❣♠❳❤s④q♦s❢✐❥①❝s⑤⑥♦✉❤s❷☎❤s❝s♠❳⑨✸❤➆❾✌➄✙➅➆⑩❬➇➪➈✙➉ ❷↕➊➌➋➎➍✬❼➆➊➌➏✂➐❱➑➒➍✬➀♠➓✯➉❹♣

➦✪✩ ✫ã★

➟✑↔

➠⑥➠⑥➦ ♣❊★

➟➌➣⑥➟

③✽✼➁❾❅✙❾✉✼✧⑤⑥⑦✐✇❣② ❧❫❛❣t ❃✧❾

❾➈●➈❧❫❛❮✽❀⑤✚❧❫❛❿❾➶➇➽➙➌➍✬❼④➀➠➢➹❷◗➊➌➓❅➑✢❿✝➍✩➙➌➍✬➀♠➊✖➋➎❺❡♣

♦❡t✂♠qt①❢❝❡❢✐⑤⑥❛❿❾ ➉✝❞①♠✵⑥✚⑤⑥❞①❛❣❤✓❅P⑤⑥❥①❼☎❢✐❛❣❤ ❆✝❛❣❢❸✚♠❳♦❡❤s❢❝❜❷❉➃✾♦❡♠❳❤❡❤q♣ ➡ ❧❫⑦❝s❢✐⑨✸⑤⑥♦❡♠⑥♣①◆✓❧❫♦s❷☎⑦❹❧❫❛❣t❿♣✠★ ➟⑥➟ ➧①❾

★ ➥ ③✽✼➁❾q✽❀♠❳②❣♦❡♠➃❝q❾❁✰✚❝❡♦s✇❣④➃❝❡✇❣♦❡❧❫⑦☎❤s⑤⑥⑦✐✇①❝s❢✐⑤⑥❛➁⑤❫⑩☎❝❡❞❣♠➈t❣❢✐❤s❝s✇❣♦❡②❣❧❫❛❣④❳♠❊t❣♠❳④❳⑤⑥✇❣❥❣⑦✐❢✐❛❣⑧P❥❣♦❡⑤⑥②❣⑦✐♠❳⑨❮⑩❺⑤⑥♦❀❢✐⑨✸❥❣⑦✐❢✐④❳❢❝r⑦✐❢✐❛❣♠q❧❫♦❑t❣❢✐❤❡④❳♦❡♠➃❝s♠➃❻ ❝s❢✐⑨✸♠▲❤✈❷①❤✈❝❡♠❳⑨✸❤➆❾❴❷✥➀➔❼➆➭✑❿➁➀➔➍✬❺⑦➄✢➦➌❺➛➍✬❻✑➓❁❺✭➄✢➀➠➜➌➋➒➙✖➏➒❶✪❼➆➊✑➭❙❻✑❺➛❺✓➉✐♣◗★ ➠ ✫➠ ★✳★ ↔☎➠⑥➦✪✩ ♣◗★ ➟⑥➟❫➥ ❾

③➋●✣❾♥◆✦♠❳❞❣♦s⑨❯❧❫❛❣❛❿❾❧➥➧➾➁❻❁⑩❬❿✝➍✩➊➌➋➒➊➌➓✯➊➌❿➁❺❀⑨↕➀➔➋➒❻➛➙➌❼❢➘✂❿➁➙✖➂✖❼➆➙➌➍✬➀♠➭➽❷↕➊➌➋➎➍✬❼➆➊➌➏❱❶✂❼➆➊✝➝✓➏❻➟➓→➴➩➥➧➾✙❻❙➊✖❼④➦❧➙✖➋➒➂❉➯❬❿➁➓❁❻✑❼④➀♠➭➛➙➌➏ ⑩⑦➏➜✎➊➌❼④➀♠➍➔➾➁➓❀❺❽♣❪❸✚⑤⑥⑦✐✇❣⑨✸♠❶★q➧

Referensi

Dokumen terkait

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume

A publication of the International Linear

A publication of the International Linear

The Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp.. Combinatorial matrix classes