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Γραμμικη Αλγεβρα

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Γραμμικη Αλγεβρα

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Page 1: Γραμμικη Αλγεβρα

������������������� ��������������� ����������������������� "!�#$� %&� '��(������������������ "!)#�� %&� '��

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Page 2: Γραμμικη Αλγεβρα

UNV W�XSYAZ [ \�] ^�_ Z ` a ` b c ^�d eX ]` f Z�] g ] h eX ] i f�b c ^ YAi _ j�[�` klZ�\b YYAi _ jAd klZ m

n c _ YA^oi X p X ` _o]f�_ Z _ g f q ^o\Aa X b \f�[�f�q XN^oc ^] ^og [ c ^ ` X�` i hoc ^r_ Z _ ` p c kl\Af�[�_ Z _ sc _ i _ YAktYA[uv^ wo] xoylz�[rq X i ] xoylz�` ^obrc X i ] X e^oq d Z ^ob�` ^obrc _ i h Z ` ^oz|{] { woa ^oboutekli a z` f Z|d YYAi _ jAf�V g X ] _�` klZ�\b YYAi _ jAd klZ m

ISBN 960-91725-0-4

Copyright (c) 2001 by N. Kadianakis and S. Karanasios.

All rights reserved. No part of this publication may be reproduced or tran-

smitted in any form or any means; electronic or mechanical, including pho-

tocopy, recording, or any information storage and retrieval system, without

permission in writing from the authors.

Page 3: Γραμμικη Αλγεβρα

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B % � íoìAò�� ô é�� C�m ø��vX Z ] xoV�mNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmÒø þ���m Êûÿ�i V � X ] z�\A` ^

Πµ×ν(K)mNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmDø ü Ê

�m þ��t] Z hoq X Z ^�co] Z V xoklZ�mNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmDø ü ü�m ü,Ý"p Z W�X ` ^]coa Z _ xoX z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mDø ü���m ���vX Z ] xoV�c _ i _ g X a YAq _ ` _�xo_ ]ù"jA_ i q ^ YAd z&mNmNmNmNm|mNmNmNm|mNmNmNm|mDø � þ�m � n \xo[ \X ] z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNmÒø � �

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�om ü n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmÒø � üC�IJ?oìAðvðtë ò�;�K|îAïlñ LAðtìñ ì é�C�C

� m ø��vX Z ] xoV�mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmDø � �� m Êûùlcoa w b \f.�vi _ q qo] xoylZ|\Ab \A` f q V ` klZ mNmNm|mNmNmNm|mNmNmNm|mNmNmNmNmÒø � Þ

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mNmNm|mNmNmNm|mNmNmNm|mNmDø Þ Ê� m þ Ý"b \` [ q _ ` _

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� m ü n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmÒø � ËAûê}ë ìAíîAïtðvìAñë òvö���*G>@?övë é�#véÞom ø��vX Z ] xoV�mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmDø �oøÞom Ê É i ] \q hozN` ^obON�] _ Z b \Aq _ ` ] xo^op�eyli ^ob�mNmNmNm|mNmNmNm|mNmNmNmNm|mNmDø � ÊÞom þ�N�] _ Z b \q _ ` ] xo^aAb c h ekli ^]DmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmDø � �Þom üP�vi _ q qo] xo^a\b Z g b _ \q ^aos"V WAi ^] \q _�b c ^ eyli klZ�mNmNm|mNmNmNmNm|mNmDø � �Þom ���vi _ q qo] xo[�_ Z X � _ i ` f \�a _ mNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNm�Ê Ë �Þom �ûUNw ] q _ xokl` [�q ^oi jA[�g ] _ Z b \q V ` klZ�` ^ob

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ø Ëom ü É q ^ YAX Z X a zN\b Z ` X ` _ YAq d Z X zN\A` ^�X c a c X g ^rm|mNmNmNmNm|mNmNmNm|mNmNmNmrÊ Þ �ø Ëm üAm ø É q ^oc _ i _ w w f w ] xohozNq X ` _ \ef q _ ` ] \Aq hoz�m|mNmNmNm|mNmNmNm|mNm�Ê Þ �ø Ëm üAm Ê�ÿ�_ i V w w f w f�q X ` _ j�^oi V�\AX�^oq ^ YAX Z X a z�\b Z ` X ` _ YAq d Z X z}mNm|m�Ê Þ �ø Ëm üAm þY�vi _ q qo] xo^aYAX klq X ` i ] xo^aAq X ` _ \Aef q _ ` ] \Aq ^a�\X|^oq ^ YAX Z X a z

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ø Ëom � n \xo[ \X ] z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNmrÊ � �éAéRZ�ì��"ðvó�ôG� � íìAò�ì4ò�ì�ë��':"ì�?oðvöA÷�Q ô �# Cø ø m øW��_ W�q hozNcoa Z _ xo_HmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNm�Ê � �ø ø m Ê�Ý"` ^] eX ] ylg X ] zNÿNa Z _ xoX z�mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNm�þ Ëoøø ø m þ�ùlj�_ i q ^ YAd zSm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNm�þ Ë �ø ø m þm øHN�] X i X p Z f \Af�\Ab \A` f q V ` klZ�mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNmrþ Ë �ø ø m þm Ê[7�c ^ w ^ Y�] \q hoz n Z ` a \` i ^oj�^ob|ÿNa Z _ xo_&mNmNm|mNmNmNm|mNmNmNmrþ Ë �ø ø m þm þY�

LUc _ i _ YA^ Z ` ^oc ^a f \f|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|m�þoø Ë

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ø ø m þm �Y�vi _ q qo] xo[ n Z X � _ i ` f \Aa _|s É i a F ^ob \_Wronski

m|mNmNmNm|m�þoø þø ø m ü n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmrþoø �

é� Òê}ë ìAíîAïtðvìAñë òvö���*G>@?övëtð��2��ïJUSñ�� ?ë òvóWI�ë íoóvð�� íö �vé�#ø Êm ø�ù"] \_ YAktY�] xoVÔmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNm�þø �ø Êm Ê�vklZ a _�g p ^�g ] _ Z b \Aq V ` klZ s É i WA^ YAylZ ] _�g ] _ Z p \q _ ` _Hm|mNmNmNm|mNmNm�þ Ê þø Êm þ É i W�^oxo_ Z ^ Z ] xod zS{oV \AX ] z&m|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNmrþ Ê �ø Êm ü$ÿNa Z _ xo_ z�X \kl` X i ] xo^op|Y�] Z ^oq d Z ^obDm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNmrþ þ Ëø Êm � É i W�^oq ^ Z _ g ] _ a ^] \

Unitary ]�^ yli ^]�m|mNmNmNm|mNmNmNmNm|mNmNmNm|mNm�þ þ Êø Êm � É i W�^ YAylZ ] ^�\Ab q c w [ i klq _ mNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|m�þ þ��ø Êm � n b ` ^o\Ab F b YAX a z'�vi _ q qo] xo^a�DX ` _ \Aef q _ ` ] \Aq ^a�mNmNmNm|mNmNmNmNm|mNm�þ ü Ëø Êm ÞS \A^oq X ` i ] xo^aoYAi _ q qo] xo^aq X ` _ \Aef q _ ` ] \qo^a)mNm|mNmNmNm|mNmNmNm|mNm�þ ü þø Êm � É i W�^ YAylZ ] ^]oq X ` _ \ef q _ ` ] \Aq ^a�m|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNmrþ ü��ø Êm ø Ë É i W�^oq ^ Z _ g ] _ a ^]�ÒX ` _ \ef q _ ` ] \q ^a�mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNm�þ ü �ø Êm ø ø n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmrþ � Ë

é��,*�ì�?ìAò�ñ V ?Aë ïlñë ò ;=��ö�ï�; � B��ø þm ø ^ _ i _ xo` f i ] \A` ] xoV�c ^o\V�YAi _ q qo] xoylZ|q X ` _ \Aef q _ ` ] \Aq ylZ)mNmNmNm|mNm�þ � þø þm Ê ^ _ i _ xo` f i ] \A` ] xoV�c ^o\V�co] Z V xoklZ�mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|m�þ � �ø þm þN�] _ YAklZ ^oc ^a f \Af�co] Z V xoklZæm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNm�þ � �ø þm þm øHN�] _ g ] xo_ \Aa _�g ] _ YAklZ ^oc ^a f \f zæmNmNmNm|mNmNmNm|mNmNmNm|mNmNm�þ � �

ø þm ü/_�X yli f q _Cayley-Hamilton

mNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNmrþ � Êø þm ��ùlw V eA] \` ^�c ^ w b ylZ b q ^�mNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNm�þ�� �ø þm � ^ _ i _ xo` f i ] \A` ] xoV�c ^o\V�co] Z V xoklZ�X ] g ] xo[ zNq ^oi j�[ z&mNmNm|mNmNmNmNm|m�þ Þ þø þm � n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmrþ � Ë

é�",`)� ñ�?ìA÷�USíAë ò Q ô2- ö�?�:JQ ô ��# Bø üAm øa�vX Z ] xoV�mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNm�þ � �ø üAm Ê n Z _ YAktYA[�\` f Z�xo_ Z ^ Z ] xo[�q ^oi j�[HmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNm�þ � �ø üAm þb_�X ` ] xoV�^oi ] \Aq d Z X zN` X ` i _ YAklZ ] xod z�q ^oi jAd z�mNm|mNmNmNm|mNmNmNmNm|mNm�üoË Êø üAm ü n \xo[ \X ] zûmNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNm�ü Ë �

Page 10: Γραμμικη Αλγεβρα

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é�B2�':"ì�?oðvöA÷�Q ô "�T Cø �om ø4UN_ q c p w X zNxo_o]X co] j�V Z X ] X z�Ê ^ob|{o_ W�q ^opouYAX Z ] xoV�mNmNm|mNmNmNm|mNmNm�üoË �ø �om Ê�UN_ q c p w X zNxo_o]ùlco] j�V Z X ] X zNq X�xod Z ` i ^�mNmNmNm|mNmNmNm|mNmNmNmNm|mNm�üAø Ëø �om þ�UN_ q c p w X zNxo_o]ùlco] j�V Z X ] X zNekli a z�xod Z ` i ^ mNmNm|mNmNmNm|mNmNmNm|mNm�üAø Þø �om ü��_ Z b \` ] xoh�Y�] Z hoq X Z ^�g ] _ Z b \Aq V ` klZ)m|mNmNmNm|mNmNmNm|mNmNmNmNm|mNm�üoÊ þø �om �X��X ` i _ YAklZ ] xo[�i a F _�coa Z _ xo_ s ÿ�^ w ] xo[�_ Z V w b \Af|mNmNm|mNmNmNmNm|mNmNmNm�ü Ê��ø �om ��U|a Z f \f�\` X i X ^op�\Aylq _ ` ^ozNq X�d Z _�\` _ WAX i h�\Af q X a ^�mNmNmNm|mNmNmNm�ü þ Ëø �om ��U|] Z f q _ ` ] xo[�` klZ�\b Z X eylZ|q d \klZ2\ ù"w _ \` ] xoh ` f ` _ ] m|mNmNmNmNm|mNm�üoþ üø �om ÞXN�] _ xoi ] ` V�g b Z _ qo] xoV�\b \` [ q _ ` _�mNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNm�ü üoËø �om ��ù�� ] \yl\AX ] zNg ] _ j�^oi ylZ�mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNm�ü ü �ø �om ø Ë n \xo[ \X ] z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm�ü ü��

é�DG1ríoì�!�!Aö�� USñ övë c �rîAò�!�ë ò�ö���d2�Aó�<�U'?oötë " B��ø �om ø4ùlb WApO4 n WAi ^] \Aq _.7�c ^ eyli klZ|m|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNm�ü�� þø �om Ê n Z _ w w ^a kl` ^]ob c h ekli ^]�mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|m�ü � �ø �om þ�UNb xowo] xo^a�7�c h ekli ^]SmNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNm�ü�� �ø �om ü n \xo[ \X ] z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm�ü � Ë

é�C2�DìAíoöAíAë ò Q ô2-�ö ?�:�Q ô "�D��ø � m ø4UN_ Z ^ Z ] xo[�q ^oi jA[�co] Z V xoklZ|klzNc i ^oz�` f�\Aed \Af�] \A^og b Z _ qoa _ z�mNm|mNm�ü�� þø � m Ê�UN_ Z ^ Z ] xod z'Ò^oi j�d z�klzNc i ^oz�` f�\ed \f�^oq ^] h ` f ` _ z mNm|mNmNmNm|m�ü � �ø �om Êm øH�Si ] YAklZ ] xo[.D^oi jA[|se_�X yli f q _

Schurm|mNmNmNm|mNmNmNmNm|m�ü�� Þ

ø � m þ�UN_ Z ^ Z ] xo[OÒ^oi j�[Jordan

m|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNm�ü�� þø �om þm ø�ùlp i X \f�xo_ Z ^ Z ] xo[ zSq ^oi jA[ zSxo_ ] { V \f z

Jordanxo_ ]ocoa Z _ xo_

^oq ^] h ` f ` _ zHmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm�ü Þ Êø �om þm Ê�ùlp i X \f�UN_ Z ^ Z ] xo[ z'Ò^oi j�[ z

Jordanq X�g ] _ YAi V q q _ ` _�mNm�üoÞ Þ

ø �om þm þ&UN_ Z ^ Z ] xo[OÒ^oi j�[Jordan

xo_ ]ù"w V eA] \A` ^rc ^ w b ylZ b q ^ mNm�ü � Þø � m üflf ` [�UN_ Z ^ Z ] xo[OD^oi jA[�m|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNm2� Ë þø �om üAm ø�ùlp i X \f.f"f ` [ zNUN_ Z ^ Z ] xo[ z3D^oi jA[ z}mNmNmNm|mNmNmNmNm|mNmNm2� Ë Þø �om üAm ÊYflf ` [�UN_ Z ^ Z ] xo[OD^oi jA[�xo_ ]ù"w V eA] \` ^�ÿ�^ w b ylZ b q ^ûmNm|mR�ø Ê

ø � m � n \xo[ \X ] z mNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm2�oø ü

Page 11: Γραμμικη Αλγεβρα

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é�A % ì�?�;�?oñ V�ðvì B�é�#ø Þm ø n Z _ w b ` ] xo[�q ^oi jA[�` f z|^oi a F ^ob \A_ z

4 7�c _ i � f�xo_ ]q ^ Z _ g ] xoh ` f ` _Fm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmR�ø �ø Þm Ê�UN_ Z ^ Z ] xo[�q ^oi j�[

Jordanm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmR� Ê Ê

ø Þm þ n w YAX { i ] xod z�g ^oq d z�mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm2� Ê �ø Þm þm ø/4 ù"Z Z ^] _�\ed \f z�sJS \^og b Z _ qoa _4m|mNmNmNm|mNmNmNm|mNmNmNmNm|mR� Ê �ø Þm þm Ê�ÿ�i V � X ] z mNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmR� Ê �ø Þm þm þYN�^oq d z$mNmNm|mNmNmNm|mNmNmNm|mNmNmNmNm|mNmNmNm|mNmNmNm|mNmNmR� Ê �

é�#,1R��ìAíoñ L�ï�� ë ô21rïtò L�ï���USí B��véZ�ë g !�ë öA÷�?oì�:e� ì B�"����î�?�� ñ L ?Aë ö B�"�D