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Page 1: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

Kef�laio �

QWROI ME ESWTERIKO

GINOMENO

��� Eisagwg�

Wc gnwst�� �nac grammik�c q�roc V or�zetai p�nw se k�poio s�ma K kai qarakth�r�zetai ap� d�o pr�xeic� thn pr�sjesh addition�

� � V � V �� V � �u� v� �� u � v �

pou e�nai eswterik� pr�xh� kai to bajmwt� � arijmhtik� pollaplasiasm� scalarmultiplication�

� � V �K �� V � �u� �� �� �u �

pou e�nai exwterik� pr�xh�

Mporo�me ep�shc na eisag�goume �na eswterik� pollaplasiasm��

h � i � V � V �� K � �u� v� �� hu � vi �me ton opo�o antistoiq�zoume �na ze�goc dianusm�twn tou V se �na arijm� tou s��matoc K� Par�deigma miac t�toiac pr�xhc e�nai to eswterik� gin�meno� innerproduct pou or�same q�ro Rn� To eswterik� gin�meno twn u� v � Rn me

u � �u�� u�� � � � � un� kai v � �v�� v�� � � � � vn� �or�zetai wc ex�c

hu � vi �nXi��

ui vi � R �

Me th bo�jeia tou eswteriko� ginom�nou� or�zontai pol� shmantik�c metrik�c �nnoiec��pwc to m�koc dian�smatoc� h ap�stash metax� d�o shme�wn� h gwn�a metax� d�odianusm�twn ka�

�Upenjum�zoume thn �parxh enallaktik�n �rwn sth bibliograf�a gia to eswterik� gin�meno stonRn� bajmwt� gin�meno �scalar product� kai gin�meno tele�ac �dot product��

Page 2: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

Sto kef�laio aut�� ja or�soume genik� thn �nnoia tou eswteriko� ginom�nou seauja�reto grammik� q�ro V kai sth sun�qeia ja eisag�goume tic �nnoiec tou m�kouckai thc orjogwni�thtac� Ja asqolhjo�me t�so me pragmatiko�c �so kai me miga�diko�c grammiko�c q�rouc� V �R� � V �C�� �Otan mia pr�tash kal�ptei kai tic d�opeript�seic genik�� h pragmatik� per�ptwsh e�nai eidik� per�ptwsh thc migadik�c�ja qrhsimopoio�me to sumbolism� V �K�� �pou K�R � C� T�loc pr�pei na epishm��noume �ti kur�wc mac endiaf�roun peperasmenodi�statoi grammiko� q�roi� �pwc oiKn kai Mm�n�K�� Merik� jewr�mata tou kefala�ou den isq�oun se apeirodi�sta�touc q�rouc� �pwc o C�K��a� b� kai aut� ja upogramm�zetai saf�c stic diatup�seictwn jewrhm�twn aut�n�

Page 3: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou �

��� H �nnoia tou eswteriko� ginom�nou

Gia l�gouc kajar� didaktiko�c ja d�soume pr�ta ton orism� tou eswteriko� gin�me�nou se pragmatik� grammik� q�ro kai sth sun�qeia to genik�tero orism� gia migadik�grammik� q�ro�

Orism�c ����

Kalo�me eswterik� gin�meno inner product � bajmwt� gin�meno scalarproduct se �na pragmatik� grammik� q�ro V �R� mia apeik�nish thc morf�c

h � i � V � V �� R

h opo�a gia k�je u� v� w � V kai � � R �qei tic ak�loujec idi�thtec A� hu � ui � �A� hu � ui � � �� u � �A� i hu� v � wi � hu � wi� hv � wi

ii h�u � vi � �hu � viA� hu � vi � hv � uiO arijm�c hu � vi kale�tai eswterik� gin�meno � bajmwt� gin�meno twn u

kai v�

Parathr�seic

�� H idi�thta A� e�nai gnwst� wc idi�thta thc jetik�thtac positivity property� thc jetik�c orisim�thtac positive deniteness� L�me �ti to eswterik� gin��meno e�nai jetik� orismno positive denite� H idi�thta A�i kale�tai idi��thta thc prosjetik�thtac additivity property en� h A�ii anaf�retai wcomogen�c idi�thta homogeneity property� Oi d�o sq�seic sthn A� gr�fontaisuqn� maz� wc ex�c

h�u � �v � wi � � hu � wi � � hv � wi � u� v� w � V � �� � � R � ����

H A� kale�tai t�te grammik� idi�thta linear property� T�loc� h A� kale�taisummetrik� idi�thta symmetric property�

�� Se poll� bibl�a� h A� gr�fetai me dipl� sunepagwg�

hu � ui � � � u � � �

To �ti �mwc h u�� sunep�getai thn hu � ui�� den e�nai apara�thto afo� aut�prok�ptei ap� thn A�� Epeid�

h�u � ui � � hu � ui � � R �

gia u�� �qoume

h�� � �i � � h� � �i � � R �� h� � �i � � h� � �i � � R ���� � �� h� � �i � � � � R ��

h� � �i � � � �� �

Page 4: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

�� Oi idi�thtec A� kai A� mporo�n na sunduasto�n wc ex�c

hu � ui � � u � V n f�g �

�� Ekt�c ap� to sumbolism� h � i� sunanto�me ep�shc sthn bibliograf�a kai �llouc��pwc oi

h j i � � � � � � j � � � � � kai �O teleuta�oc sumbolism�c dhl� h tele�a � kal� e�nai na apofe�getai gia namhn up�rqei k�ndunoc s�gqushc me to sun�jh � to bajmwt� pollaplasiasm��

�� �Ena eswterik� gin�meno pou gia u � V ikanopoie� th sunj�kh

hu � vi � � v � V �� u � �

kale�tai mh ekfulismno non�degenerate�

To eswterik� gin�meno or�zetai an�loga kai se migadik� grammik� q�ro� V �C��me m�nh diafor� to �ti h summetrik� idi�thta A� antikaj�statai ap� th suzug�summetrik� idi�thta conjugate symmetric property

hu � vi � hv � ui � u� v � V �C� �

E�nai faner� �ti h suzug�c summetrik� idi�thta perilamb�nei wc eidik� per�ptwsh thsummetrik� idi�thta se pragmatiko�c grammiko�c q�rouc� Sunep�c� o orism�c pouakolouje� apotele� gen�keush tou Orismo� ������

Orism�c �����

Kalo�me eswterik� gin�meno � bajmwt� gin�meno � ermitian� gin�menoHermitian product � ak�ma monadia�o eswterik� gin�meno unitary innerproduct se �na migadik� grammik� q�ro V �C� mia apeik�nish thc morf�c

h � i � V � V �� C

h opo�a gia k�je u� v� w � V kai � � C �qei tic ak�loujec idi�thtec A� hu � ui � �A� hu � ui � � �� u � �A� i hu� v � wi � hu � wi� hv � wi

ii h�u � vi � �hu � viA� hu � vi � hv � uiO arijm�c hu � vi kale�tai eswterik� gin�meno � bajmwt� gin�meno � ermi�

tian� gin�meno � ak�ma monadia�o eswterik� gin�meno twn u kai v�

Page 5: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou �

Orism�c �����

a �Enac grammik�c q�roc V �K� efodiasm�noc me eswterik� gin�meno h� � �il�getai q�roc me eswterik� gin�meno � q�roc eswteriko ginomnou in�ner product space kai sumbol�zetai me �V� h � i��

�Enac pragmatik�c grammik�c q�roc efodiasm�noc me eswterik� gin�menokale�tai pragmatik�c q�roc me eswterik� gin�menoa real inner productspace�

�Enac migadik�c grammik�c q�roc efodiasm�noc me eswterik� gin�menokale�tai migadik�c q�roc me eswterik� gin�meno complex inner productspace�b �Enac peperasmenodi�statoc pragmatik�c q�roc me eswterik� gin�menokale�tai eukle�deioc q�rocb Euclidean space�

�Enac peperasmenodi�statoc migadik�c q�roc me eswterik� gin�meno kale��tai monadia�oc q�roc unitary space � ermitian�c q�roc Hermitian space� migadik�c eukle�deioc q�roc�

aO orism�c aut�c e�nai sqed�n peritt�c sta ellhnik� Ton d�noume �mwc gia l�gouc plhr�thtac maz� me ton ant�stoiqo agglik� �ro�

bProsoq�� T�so sthn ellhnik� �so kai sthn agglik� bibliograf�a� oi orismo� twn eukle�deiwn kai twn monadia�wn q�rwn d�nontai suqn qwr�c thn apa�thsh oi q�roi auto� na e�naipeperasmenodistatoi�

Sun�jh eswterik� gin�mena se grammiko c q�roucTa eswterik� gin�mena pou ja oristo�n pio k�tw anaf�rontai wc ta sun�jh esw�

terik� gin�mena standard inner products stouc ant�stoiqouc grammiko�c q�rouc�

�� Q�roc Rn

�Estw ta dian�smata u��u�� u�� � � � � un� kai v��v�� v�� � � � � vn� tou Rn� To s�nh�jec eswterik� gin�meno or�zetai wc ex�c

hu � vi �nXi��

ui vi � ����

An jewr�soume ta dian�smata tou Rn wc n � � p�nakec� mporo�me ep�shc nagr�youme

hu � vi �nXi��

ui vi � uTv � vTu � ����

H ap�deixh �ti h pio p�nw pr�xh ikanopoie� tic sunj�kec tou Orismo� ����� e�naiarket� apl�� S�mfwna me ton Orism� ������ o Rn efodiasm�noc me to s�nhjeceswterik� gin�meno e�nai eukle�deioc q�roc di�stashc n� Sthn agglik� ku�r�wc bibliograf�a anaf�retai kai wc eukle�deioc n�q�roc Euclidean n�space� ak�ma wc kartesian�c q�roc Cartesian space� To s�nhjec eswterik� gin��meno � suqn� kale�tai eukle�deio gin�meno�

Ston R� to eswterik� gin�meno taut�zetai me to sun�jh pollaplasiasm�

hx � yi � xy � x� y � R �

Page 6: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

�� Q�roc Cn

To s�nhjec eswterik� gin�meno ston Cn or�zetai wc ex�c

hu � vi �nXi��

ui vi � u� v � Cn � ����

Jewr�ntac ta dian�smata tou Cn wc n � � p�nakec� �qoume

hu � vi �nXi��

ui vi � v�u � ����

�pou v� o anastrofosuzug�c tou v� v��vT � O Orism�c ����� epalhje�jetaie�kola� O �Cn� h� � �i� e�nai monadia�oc q�roc di�stashc n�

To eswterik� gin�meno ston C taut�zetai me ton pollaplasiasm� migadiko�arijmo� me to suzug� en�c �llou

hz� � z�i � z� z� � z�� z� � C �

�� Q�roc Mm�n�R�Sto q�ro twn pragmatik�n m� n p�nakwn� h sq�sh

hA � Bi � tr�BTA� � A�B � Mm�n�R� ����

or�zei �na eswterik� gin�meno� Ap� ton orism� tou pollaplasiasmo� pin�kwn�br�skoume �ti

hA � Bi � tr�BTA� � tr

�mXk��

bki akj

�n�n

�mXi��

nXj��

aij bij � ���

H epal�jeush tou Orismo� ����� e�nai apl� kai af�netai san �skhsh�O �Mm�n�R�� h� � �i� e�nai eukle�deioc q�roc di�stashc mn�

�� Q�roc Mm�n�C�H sq�sh

hA � Bi � tr�B�A� �mXi��

nXj��

aij bij � A�B � Mm�n�C� � ����

�pou B� o anastrofosuzug�c tou B� or�zei to s�nhjec eswterik� gin�meno stonMm�n�C�� H epal�jeush tou Orismo� ����� e�nai apl� kai af�netai san �skhsh�O �Mm�n�C�� h� � �i� e�nai monadia�oc q�roc di�stashc mn�

�� Q�roc C�R��a� b�Ston pragmatik� grammik� q�ro C�R��a� b� twn pragmatik�n sunart�sewn poue�nai suneqe�c sto �a� b�� h sq�sh

hf � gi �Z b

af�t� g�t� dt � f� g � C�R��a� b� �����

or�zei �na eswterik� gin�meno� Parathro�me �ti

hf � fi �Z b

af��t� dt � � �

Page 7: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou �

op�te h sunj�kh A� tou Orismo� ����� ikanopoie�tai�An

hf � fi �Z b

af��t� dt � � �

t�te epeid� h f e�nai suneq�c sto �a� b� kai f��x� � � gia x � �a� b�� h f e�naianagkastik� h mhdenik� sun�rthsh� �pwc apaite� h sunj�kh A�� Oi idi�thtecA� kai A� apodeikn�ontai e�kola me thn bo�jeia twn idiot�twn tou orism�nouoloklhr�matoc�

�� Q�roc C�C��a� b�To s�nhjec eswterik� gin�meno sto migadik� grammik� q�ro C�C��a� b� or�zetaiap� thn

hf � gi �Z b

af�t� g�t� dt � f� g � C�C��a� b� � �����

Par�deigma Gia to eswterik� gin�meno twn dianusm�twn u���� � ����� kai v���� � �� �� tou R�

pa�rnoume

hu � vi � h��� � ����� � ��� � �� ��i � � � � � � � � � � � ���� � � � � �

Par�deigma ��Eqoume �dh anaf�rei �ti s��na grammik� q�ro mporo�me na or�soume periss�teraap� �na eswterik� gin�mena� Ja de�xoume �ti h pr�xh

hu � vi � u�v� � u�v� � u�v� � u�v�

or�zei �na eswterik� gin�meno ston R��

A� hu � ui � u�� � u�u� � u�� � u�� � �u� � u��� � �A� hu � ui � � �� u� � u� � � �� u � �A� Gia u� v� w � R� pa�rnoume

hu� v � wi � �u� � v��w� � �u� � v��w� � �u� � v��w� � �u� � v��w� ��hu�v�wi � � u�w��u�w��u�w��u�w���� v�w��v�w��v�w��v�w�� � hu�wi�hv� wiGia � � R �qoume ep�shc

h�u � vi � ��u��v� � ��u��v� � ��u��v� � ��u��v� ��h�u � vi � � � u�v� � u�v� � u�v� � u�v�� � � hu � vi

A� hu � vi � v�u� � v�u� � v�u� � v�u� � hv � uiAfo� ikanopoie� tic sunj�kec A��A� tou Orismo� ������ h pr�xh h� � �i or�zei �naeswterik� gin�meno ston R��

Par�deigma �To eswterik� gin�meno tou prohgo�menou parade�gmatoc mporo�me na to gr�youmewc ex�c

hu � vi � �u� u��

� ��

�� �

� �v�v�

�� uTAv �

Page 8: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

�pou

A �

� ��

�� �

��

Ep�shc� mia �llh parat�rhsh e�nai �ti mporo�me na gr�youme to s�nhjec eswterik�gin�meno ston Rn wc ex�c

hu � vi � uTIv � u� v � Rn � ��� �

Me aform� aut�c tic parathr�seic� ac jewr�soume thn apeik�nish

hu � vi � uTAv � u� v � Rn � �����

�pou A � Mn�n�R�� Ja bro�me tic idi�thtec pou pr�pei na �qei o A gia na e�nai hpio p�nw apeik�nish eswterik� gin�meno ston Rn�

H A� ikanopoie�tai ap� k�je n� n p�naka A afo� gia k�je u� v� w � Rn

hu� v � wi � �u� v�T Aw � �uT � vT �Aw � uTAw � vTAw � hu � wi � hv � wikai gia k�je � � R

h�u � vi � ��u�T Av � � uTAv � � hu � vi �Gia na ikanopoie�tai h A� pr�pei na isq�ei

hu � vi � hv � ui � uTAv � vTAu �

Epeid�uTAv � �uTAv�T � vT AT �uT �T � vT ATu �

h A� ikanopoie�tai an o A e�nai summetrik�c� an dhlad�AT�A� Gia na ikanopoio�ntaioi A� kai A� arke� na isq�ei

uTAu � � u �� � � �����

�Opwc ja do�me sth sun�qeia tou kefala�ou� p�nakec pou ikanopoio�n thn pio p�nwsunj�kh kalo�ntai jetik� orismnoi�

Me an�logo tr�po� br�skoume �ti h apeik�nish

hu � vi � u�Av � u� v � Cn � �����

e�nai eswterik� gin�meno ston Cn an o A � Mn�n�C� e�nai ermitian�c A��A kai

u�Au � � u �� � � �����

Par�deigma �Gia to eswterik� gin�meno twn dianusm�twn u���� � � i� kai v�� � �i� i� tou C�

�qoume

hu� vi � h��� ��i�� � ��i� i�i � �� � �i� � ���i�i � �� ��i�� ���i���i� � ��i�

Page 9: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou

Par�deigma ��Estw w � C�R��a� b� mia sun�rthsh me thn idi�thta

w�x� � � x � �a� b� �

H sq�sh

hf � gi �Z b

aw�t� f�t� g�t� dt � f� g � C�R��a� b� �����

or�zei �na eswterik� gin�meno ston C�R��a� b�� H w�x� kale�tai sun�rthsh b�roucweight function tou eswteriko� ginom�nou�

Par�deigma ��Estw V �nac peperasmenodi�statoc grammik�c q�roc kai A�fw�� w�� � � � � wng miadiatetagm�nh b�sh tou� An u� v � V me

u � a�w� � a�w� � � � � � anwn �

kaiv � b�w� � b�w� � � � � � bnwn �

t�te h pr�xh

hu � vi �nXi��

ai bi ����

or�zei �na eswterik� gin�meno ston V � H ap�deixh e�nai apl�� E�nai ep�shc faner�h analog�a me to s�nhjec eswterik� gin�meno ston Rn� H ���� e�nai apl�c toeswterik� gin�meno twn dianusm�twn twn suntetagm�nwn twn u kai v wc proc thb�sh A�

Par�deigma ��Estw Pn o q�roc twn pragmatik�n poluwn�mwn bajmo� mikr�terou � �sou me n� An

p�x� � a� � a� x � � � � � an xn kai q�x� � b� � b� x � � � � � bn x

n �

t�te apodeikn�etai e�kola �ti h sq�sh

hp � qi �nXi��

ai bi �����

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hfaig � fbigi ��Xi��

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Page 10: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

Epeid� to pio p�nw �jroisma sugkl�nei apol�twc� to eswterik� gin�meno e�nai kal�corism�no�

H pr�tash pou akolouje� sunoy�zei tic basik�c idi�thtec tou eswteriko� ginom�nou�

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S��na q�ro me eswterik� gin�meno �V �K�� h� � �i� isq�oun ta ex�c i hu � v � wi � hu � vi � hu � wi � u� v� w � V �ii hu � �vi � � hu � vi � u� v � V � � � K�iii hu � �v � �wi � � hu � vi � � hu � wi � u� v� w � V � �� � � K�iv hu � �i � h� � ui � � � u � V �v h� � �i � ��vi �

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hu � �vi � h�v � ui � � hv � ui � � hv � ui � � hu � vi �iii Prok�ptei apl� ap� tic i kai ii�iv hu � �i�hu � �ui� �hu � ui��� �Omoia br�skoume �ti h� � ui���Enallaktik� ap�deixh� Ap� thn i �qoume

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v Prok�ptei am�swc ap� thn iv j�tontac u��� Shmei�noume �ti apode�xame �dhthn h� � �i�� sthn per�ptwsh pragmatiko� grammiko� q�rou me eswterik� gin�meno�vi �Amesh gen�keush thc grammik�c idi�thtac A��

Page 11: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou ��

vii �Amesh gen�keush thc iii�viii Prok�ptei apl� ap� tic vi kai vii� �

Ap� th suzug� summetrik� idi�thta A� Orism�c ����� parathro�me �ti gia k�jeu � �V � h� � �i�

hu � ui � hu � uikai epom�nwc hu � ui � R� Aut� �llwste uponoe�tai ap� th sunj�kh jetik�c orisim��thtac A� tou Orismo� ������ Gia to mh arnhtik� arijm� hu � ui d�noume to ak�loujoorism��

Orism�c �����

�Estw �V �K�� h� � �i� q�roc me eswterik� gin�meno� Kalo�me n�rma norm� mtro � m�koc length thn apeik�nish jj � jj � V �� R pou or�zetai ap� thsq�sh

jjujj � hu � ui��� � u � V � �� ��

L�me �ti h n�rma jj � jj par�getai ap� to eswterik� gin�meno h� � �i�

Parathr�seic

�� Ant� tou jj � jj� sth bibliograf�a qrhsimopoie�tai ep�shc kai o sumbolism�c j � j�Ep�shc� sta ellhnik� sunanto�me touc �rouc norm kai st�jmh�

�� �Opwc ja do�me sto ed�fio ���� den e�nai apara�thto o V na e�nai efodiasm�nocme eswterik� gin�meno gia na or�soume mia n�rma�

Sth sun�qeia paraj�toume tic n�rmec pou prok�ptoun ap� ta sun�jh eswterik�gin�mena pou or�same sthn arq� tou edaf�ou� Aut�c e�nai ep�shc sugkentrwm�necston P�naka ����

Sun�jeic n�rmec se grammiko c q�rouc

�� Q�roc Rn

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nXi��

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jjujj �qu�� � u�� � u��

parist�nei thn ap�stash tou shme�ou u��u�� u�� u�� ap� thn arq� en�c orjog��niou sust�matoc suntetagm�nwn �� isod�nama� to m�koc tou dian�smatoc u� Hn�rma ���� e�nai mia gen�keush thc �nnoiac tou m�kouc ston Rn�

Page 12: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

�� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

P�nakac �� Sun�jh eswterik� gin�mena kai ant�stoiqec n�rmec�

�� Rn hu � vi �nXi��

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eidik c peript�seic twn orism�n stouc ant�stoiqouc migadiko�c q�rouc Cn�Mm�n�C� kai C�C� a� b��

Page 13: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou ��

Ston R� h n�rma taut�zetai me thn ap�luth tim�

jjxjj � hx � xi��� �px� � jxj � x � R �

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�� Q�roc Cn

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qu�� � �u� � u��� �

Page 14: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

�� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

Qrhsimopoi�ntac aut� th n�rma br�skoume gia to u����� �

jjujj �q�� � ����� �

p�� �

x

y

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jjujj �q� u�� � u��

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Sq�ma ��� O monadia�oc k�kloc pou or�zetai ap� th n�rma jjujj�q� u�� � u��

ston R��

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hu � vi � � u�v� � u�v�

or�zei �na eswterik� gin�meno ston R��b Na breje� h n�rma pou par�getai ap� to pio p�nw eswterik� gin�meno�g Sqedi�ste ton monadia�o k�klo pou or�zetai ap� thn pio p�nw n�rma�

a Ja de�xoume �ti isq�oun oi idi�thtec A��A� tou Orismo� ������A� hu � ui � � u�� � u�� � �A� hu � ui � � �� u� � u� � � �� u � �A� Gia u� v� w � R� pa�rnoume

hu�v�wi � ��u��v��w���u��v��w� � ��u�w��u�w�����v�w��v�w�� � hu�wi�hv� wi�Gia � � R �qoume ep�shc

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jjujj � hu � ui��� �q� u�� � u�� �

g J�tontac jjujj�� �qoume� u�� � u�� � � �

O monadia�oc k�kloc antistoiqe� se �lleiyh me �xonec ��p� kai � bl� Sq�ma ����

Page 15: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H �nnoia tou eswteriko� ginom�nou ��

Par�deigma �Gia th sun�jh n�rma tou u�� � �i� �� i� � C�� �qoume

jjujj �qj � �ij� � j�� ij� �

p�� � � �

p� �

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A �

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f�x� � ei mx � C�C���� �� �

�pou m ak�raioc� �qoume

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Page 16: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

�� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

��� H anisthta Cauchy�Schwarz

H anis�thta Cauchy�Schwarz e�nai ap� tic shmantik�terec anis�thtec kai e�nai pol�qr�simh se pollo�c kl�douc twn majhmatik�n� Anaf�retai kai wc anis�thta touSchwarz� kur�wc sth Germanik� bibliograf�a� wc anis�thta tou Cauchy�� kur�wc sthGallik� bibliograf�a� kai wc anis�thta tou Bunyakowsky� kur�wc sth Rwsik� biblio�graf�a�

Pr�tash ���� �Anis�thta Cauchy�Schwarz�

�Estw �V� h � i� �nac q�roc me eswterik� gin�meno� Gia k�je u� v � V isq�ei

hu � vi� hu � ui hv � vi �� �

� isod�namajhu � vij jjujj jjvjj � �� ��

H is�thta isq�ei �tan kai m�no �tan ta u kai v e�nai grammik�c exarthm�na�

Ap�deixhGia v��� h � isq�ei san is�thta afo� hu��i�� kai jjujj jjvjj�jjujj ���� An t�rav �� �� jewro�me to eswterik� gin�meno

A �Dujjvjj� � hu � viv � ujjvjj� � hu � viv

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l�gw thc jetik�thtac tou eswteriko� ginom�nou� Gia to A �qoume diadoqik�

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� jjvjj� jjujj� � jjvjj� hu � vi hu � vi� jjvjj� jjujj� � jjvjj� jhu � vij�� jjvjj�

�jjujj� jjvjj� � jhu � vij�

��

�O Glloc majhmatik�c Augustin�Louis Cauchy ���������� sugkatal getai stouc megal�teroucmajhmatiko�c �lwn twn epoq�n� �Egraye � bibl�a kai periss�tera ap� ��� rjra me j mata ap��louctouc kldouc twn majhmatik�n� �Ejese tic bseic thc s�gqronhc migadik�c anlushc� kai e�qe shmantik� sumbol� sthn kumatik�� th jewr�a elastik�thtac� th jewr�a orizous�n� th jewr�a idiotim�n�th mel th sun�jwn kai merik�n diaforik�n exis�sewn� th jewr�a twn omdwn metaj sewn� kai thjemel�wsh tou majhmatiko� logismo�� �Orise tic akolouj�ec pou f roun to �nom tou� Eis�gage ton�ro or�zousa �determinant� kai to sumbolism� diplo� de�kth gia ta stoiqe�a pinkwn� Anakluye tonupologism� or�zousac wc anptugma algebrik�n sumplhrwmtwn� Parous�ase pol� pio prin ������ap� touc Bunyakovsky ������ kai Schwarz ������ thn anis�thta Cauchy�Schwarz kai �rise pr�toc tom�koc kai to embad�n wc oloklhr�mata�

Page 17: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H anis�thta Cauchy�Schwarz ��

Epeid� A � � kai v �� ��

jhu � vij� jjujj� jjvjj� �� jhu � vij jjujj jjvjj �Ja de�xoume t�ra �ti h is�thta sthn i isq�ei �tan kai m�no �tan ta u kai v e�naigrammik�c exarthm�na� E�nai profan�c �ti an v �� � kai A��� t�te giat��

ujjvjj� � hu � vi v � � �� u �hu � vijjvjj� v �

Afo� to u e�nai bajmwt� pollapl�sio tou v� ta u kai v e�nai grammik�c exarthm�na�Ant�strofa� an u��v� � � C� t�te

ujjvjj��hu� viv � �v jjvjj�� h�v � viv � �v jjvjj�� �hv � viv � �v jjvjj�� �v jjvjj� � � �

kai �tsi A�h� � �i��� �

Gia mia diaforetik� ap�deixh thc anis�thtac Cauchy�Schwarz� bl� Pr�blhma ����

Par�deigma Jewro�me to q�ro Rn efodiasm�no me to s�nhjec eswterik� gin�meno� An

u � �a�� a�� � � � � an� kai v � �b�� b�� � � � � bn� � Rn �

t�te ap� thn anis�thta Cauchy�Schwarz br�skoume th gnwst� anis�thta�nXi��

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Ston P�naka ��� fa�nontai oi anis�thtec pou prok�ptoun me efarmog� thc anis��thtac Cauchy�Schwarz stouc kuri�terouc q�rouc pou mac apasqolo�n me to s�nhjeceswterik� gin�meno�

Page 18: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

�� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

P�nakac ��� Anis�thtec pou prok�ptoun me efarmog� thc anis�thtac

Cauchy�Schwarz me to s�nhjec eswterik� gin�meno stouc ant�stoiqouc q�rouc�

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Page 19: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� H anis�thta Cauchy�Schwarz �

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H pr�tash pou akolouje� sunoy�zei tic basik�c idi�thtec thc n�rmac�

Pr�tash �����

�Estw �V� h � i� �nac q�roc me eswterik� gin�meno kai h n�rma jj � jj�h� � �i����Gia k�je u� v � V kai � � C isq�oun ta ak�louja B� jjujj � � jetik�thtaB� jjujj � � �� u � ��B� jj�ujj � j�j jjujj omogneia � omogen�c idi�thta�B� Trigwnik� anis�thta triangle inequality

jju� vjj jjujj � jjvjj � �����

Ap�deixhB�B� Prok�ptoun e�kola ap� ton orism� thc n�rmac kai tic idi�thtec tou eswte�riko� ginom�nou�B�

jj�ujj� � h�u � �ui � �� hu � ui � j�j� jjujj� �� jj�ujj � j�j jjujj �B�

jju � vjj� � hu � v � u� vi � hu � ui � hu � vi � hv � ui � hv � vi� jjujj� � hu � vi � hu � vi � jjvjj� � jjujj� � Rehu � vi � jjvjj� jjujj� � jhu � vij � jjvjj� �

Qrhsimopoi�same basik�c idi�thtec twn migadik�n�� Me thn anis�thta Cauchy�Schwarz pa�rnoume

jju � vjj� jjujj� � jjujjjjvjj � jjvjj� � �jjujj � jjvjj�� ��jju � vjj jjujj � jjvjj �

�An z � C� t�te z � z��Rez kai Rez � jzj�

Page 20: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

� Kef�laio �� QWROI ME ESWTERIKO GINOMENO

E�nai faner� �ti h is�thta isq�ei an u�� � v��� Isq�ei ep�shc an v��u me � ���op�te

jju � vjj � jju � �ujj � jj�� � �� ujj � j� � �j jjujj � � jjujj � � jjujj ��jju � vjj � jjujj � jj�ujj � jjujj � jjvjj �

Parathro�me �tijj � ujj � j � �j jjujj � jjujj �

����

���JJJJ�

��������

u

vu� v jju � vjj jjujj � jjvjj

Sq�ma ���� H trigwnik� anis�thta ston R��

H anis�thta ���� onom�zetai trigwnik� anis�thta giat� ston R� aut� an�getaisth gnwst� pr�tash pou l�ei �ti to m�koc miac pleur�c en�c trig�nou e�nai mikr�teroap� to �jroisma twn mhk�n twn d�o �llwn pleur�n Bl� Sq�ma ����

Ap� thn trigwnik� anis�thta prok�ptoun e�kola h

jju � vjj jju � wjj � jjv � wjj � u� v� w � V �����

kaj�c kai h ant�strofh trigwnik� anis�thta� reverse triangle inequality

j jjujj � jjvjj j jju � vjj � u� v � V � �����

H ap�deixh twn ���� kai ���� af�netai wc �skhsh�

Pr�tash ����� N�moc tou parallhlogr�mmou �parallelogram law�

An �V� h � i� �nac q�roc me eswterik� gin�meno kai u� v � V � t�te

jju� vjj� � jju� vjj� � �jjujj� � jjvjj�

�� �����

Ap�deixhGia k�je u� v � V �qoume

jju� vjj� � hu � v � u� vi � jjujj� � Rehu � vi � jjvjj�jju� vjj� � hu � v � u� vi � jjujj� � Rehu � vi � jjvjj� �

Prosj�tontac tic pio p�nw sq�seic pa�rnoume thn ����� �

Ston P�naka ��� sugkentr�same basik�c anis�thtec kai is�thtec pou isq�oun s��naq�ro me eswterik� gin�meno kai thn ant�stoiqh n�rma�

�H ant�strofh trigwnik� anis�thta anaf retai ep�shc kai wc trigwnik� anis�thta proc ta k�tw�

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���� H anis�thta Cauchy�Schwarz ��

P�nakac ��� Basikc idi�thtec thc n�rmac jj � jj�h� � �i��� se q�rouc me eswterik�

gin�meno�

�V� h � i� kai jj � jj � h� � �i��� u� v� w � V � � � K

B Jetik�thta jjujj � �

B� jjujj � � � u � �

B� Omogen�c idi�thta jj�ujj � j�j jjujj

B� Trigwnik� anis�thta jju � vjj jjujj � jjvjj

jju � vjj jju � wjj � jjv � wjj

Ant�strofh trigwnik� anis�thta j jjujj � jjvjj j jju � vjj

Anis�thta Cauchy�Schwarz jhu � vij jjujj jjvjj

N�moc tou parallhlogr�mmou jju� vjj� � jju� vjj� � �jjujj� � jjvjj��

Page 22: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

��� Q�roi me n�rma

S��na grammik� q�ro den e�nai apara�thto na �qoume or�sei eswterik� gin�meno giana or�soume mia n�rma� �Eqoume ton pio k�tw orism�

Orism�c �����

Kalome n�rma �norm� p�nw ston grammik� q�ro V �K� mia apeik�nish thcmorf c jj � jj � V �� R h opo�a� gia k�je u� v � V kai � � K� ikanopoie� ticak�loujec sunj kec�B�� jjujj � � �jetik�thta���B�� jjujj � � �� u � ���B�� jj�ujj � j�j jjujj �omogen�c idi�thta���B�� jju� vjj � jjujj� jjvjj �trigwnik� anis�thta���Enac grammik�c q�roc V �K� efodiasm�noc me mia n�rma jj � jj l�getai q�rocme n�rma �normed space� kai sumbol�zetai me �V �K�� jj � jj��

Parathr�seic

�� Sthn ellhnik bibliograf�a� gia to q�ro me n�rma qrhsimopoiontai ep�shc oi�roi �nnormoc� normarism�noc stajmht�c q�roc�

�� Se poll� bibl�a� h B� gr�fetai wc ex c

jjujj � � �� u � � �

�Omwc to �ti jj�jj�� den e�nai apara�thto afo prokptei ap� thn B�

jj��jj � j�j jj�jj � � K �� jj�jj � j�j jj�jj � � K ���� � j�j� jj�jj � � � � K �� jj�jj � � �

�� H sunj kh B� mpore� na paralhfje� afo prokptei ap� tic B��B�� �Opwcde�xame pio p�nw jj�jj�� kai �tsi

jju� ��u�jj � � �

Me thn trigwnik anis�thta B� �qoume

jjujj � jj � ujj � �

kai me thn omogen idi�thta B��

jjujj � j � �j jjujj � � �� � jjujj � � �� jjujj � � �

�� �Opwc e�dame sto ed�fio ���� ap� thn trigwnik anis�thta prokptei ekola henallaktik thc morf

jju � vjj � jju � wjj � jjv � wjj � u� v� w � V ���

kaj�c kai h ant�strofh trigwnik anis�thta

j jjujj � jjvjj j � jju � vjj � u� v � V � ����

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���� Q�roi me n�rma ��

�� O prosektik�c anagn�sthc ja �qei parathr sei �ti oi idi�thtec B��B� e�naiakrib�c aut�c pou apode�xame sthn Pr�tash ������ �Enac q�roc eswterikoginom�nou �V� h � i� e�nai eidik per�ptwsh q�rou me n�rma afo sto eswterik�gin�meno antistoiqe� h n�rma

jj � jj � h� � �i��� �To ant�strofo den isqei p�nta� �Enac q�roc me n�rma �V� jj � jj� ston opo�oisqei h idi�thta tou parallhlogr�mmou e�nai q�roc eswteriko ginom�nou afooi pr�xeic

hu� vi ��

�jju � vjj� � jju � vjj�

�an V�V �R�� kai

hu� vi ��

hjju � vjj� � jju � vjj� � i

�jju � ivjj� � jju � ivjj�

� i

an V�V �C�� apotelon eswterik� gin�mena stouc ant�stoiqouc q�rouc��

�� An mia apeik�nish ikanopoie� tic idi�thtec B�� B� kai B� all� �qi kat� an�gkhth B�� t�te aut onom�zetai hmin�rma �seminorm�� Mia hmin�rma epitr�pei thnparxh kai �llwn dianusm�twn u � V � ekt�c tou mhdeniko� pou ikanopoion thnjjujj���

�Opwc ja dome pio k�tw� s��na q�ro mporome na or�soume poll�c diaforetik�cn�rmec tic opo�ec diakr�noume qrhsimopoi�ntac diaforetikoc upode�ktec� Up�rqoundhlad pollo� tr�poi gia na metr soume to m koc en�c diansmatoc ap� touc opo��ouc epil�goume kat� per�stash ton pio bolik�� Sth sun�qeia� ja or�soume n�rmecdianusm�twn �ston Kn�� pin�kwn �ston Mm�n� kai sunart sewn �ston C�K��a� b���

Ston q�ro Kn mporome na or�soume thn pn�rma gia � � p �

jjujjp �

�nXi��

juijp���p

� u � Kn � � ��

Endiaf�rousec e�nai oi eidik�c peript�seic

jjujj� �nXi��

juij � ��

jjujj� �

�nXi��

juij�����

� ��

jjujj� � max��i�n

juij � � ��

Oi n�rmec jj � jj�� jj � jj� kai jj � jj� kalontai �nan�rma� don�rma kai �peiron�rma� ant�stoiqa� Anaf�rontai ep�shc wc l�� l� kai l� � L�� L� kai L�� n�rmec�

�H ap�deixh af�netai san �skhsh� Prosoq�� den e�nai tetrimm�nh�

Page 24: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

(1, 0)(1, 0)

(1, 1)

(−1, −1)

(1, 0)

S� S� S�

Sq�ma ���� Oi monadia�oi k�kloi S�� S� kai S��

Apodeiknetai ekola �ti oi n�rmec aut�c ikanopoion tic apait seic tou Orismo������ Eidik� gia th ��n�rma parathrome �ti e�nai h n�rma tou Kn pou or�zetai ap�to snhjec eswterik� gin�meno

jjujj� � hu � ui � �uTu���� � u � Rn � �

kaijjujj� � hu � ui � �u�u���� � u � Cn � � ��

H ��n�rma kale�tai ep�shc ajroistik� n�rma� E�nai pol qr simh se probl mataektimhtik c sth Statistik kaj�c kai sth jewr�a prosegg�sewn� H ��n�rma kale�taiep�shc eukle deia n�rma kai sumbol�zetai kai me jj�jjE� Aut �qei apodeiqje� qr simhse probl mata beltistopo�hshc� H �n�rma kale�tai ep�shc n�rma meg stou �maxnorm� ak�ma omoi�morfh n�rma �uniform norm� kai e�nai qr simh kur�wc gia thnap�deixh jewrhm�twn sgklishc� sth jewr�a stoqastik�n anel�xewn kai th sgklishalgor�jmwn�

Sth genik per�ptwsh �� � p � � p ���� h ap�deixh �ti h jj � jjp apotele� n�rmae�nai mh tetrimm�nh� Aut� den mac enoqle� idia�tera afo oi peript�seic pou macendiaf�roun e�nai oi p��� � kai � Mporome na dome mia gewmetrik ermhne�a giatic treic aut�c n�rmec� an jewr soume touc monadia�ouc k�klouc

Sp �nx � R� � jjxjjp � �

o�

Oi monadia�oi kkloi S�� S� ka S� fa�nontai sto Sq ma ����

Par�deigmaGia to u���� �� �� ��� � R� pa�rnoume

jjujj� � j�j � j�j � � � j � �j �

jjujj� �p� � � � � � �

p�

jjujj� � maxf�� �� �� �g � �

Page 25: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� Q�roi me n�rma ��

�Ena klasik� apot�lesma gia thn p�n�rma ston Rn e�nai h anis�thta H�older�

j uTv j � jjujjp jjujjq � �

p�

q� � � � ��

H anis�thta Cauchy�Schwarz e�nai profan�c mia shmantik eidik per�ptwsh thc ani�s�thtac Holder

j uTv j � jjujj� jjujj� � � ��

D�noume qwr�c ap�deixh to pio k�tw shmantik� je�rhma��

Pr�tash �����

�Olec oi n�rmec ston Rn e�nai isodnamec �equivalent�� dhl� an jj � jj� kai jj � jj�e�nai n�rmec ston Rn� t�te gia k�je u � Rn up�rqoun jetik�c stajer�c c� kaic� t�toiec �ste

c� jjujj� � jjujj� � c� jjujj� � � �

Gia par�deigma� an u � Rn �qoume

jjujj� � jjujj� �pn jjujj� � � ��

jjujj� � jjujj� �pn jjujj� � ����

jjujj� � jjujj� � n jjujj� � ����

H ap�deixh twn pio p�nw sq�sewn af netai wc �skhsh�

Oi n�rmec p�naka e�nai pol qr simec sthn arijmhtik grammik �lgebra� Afoo Mm�n e�nai is�morfoc me ton Kmn� o orism�c thc n�rmac p�naka pr�pei na e�naiisodnamoc me ton orism� thc n�rmac diansmatoc� Oi pio qr simec n�rmec p�nakae�nai h n�rma Frobenius� �Frobenius norm��

jjAjjF �

�� mXi��

nXj��

jaijj������

� A � Mm�n�K� � ����

h opo�a e�nai h n�rma pou or�zetai ap� to snhjec eswterik� gin�meno ston Mm�n�kai h p�n�rma�

jjAjjp � supu���

jjAujjpjjujjp � A � Mm�n�K� kai u � Kn � ����

�O German�c algebrist�cOtto Ludwig H�older �� � ��� asqol�jhke kur�wc me th jewr�a om�dwn�Anak�luye th gnwst� anis�thta to ��� kai eis�gage thn �nnoia thc om�dac parag�ntwn factorgroup� to �� �

�Gia thn ap�deixh apodeikn�oume pr�ta �ti h n�rma jj � jj ston Kn e�nai suneq�c sun�rthsh��H n�rma Frobenius anaf�retai merik�c for�c kai wc eukle�deia n�rma p�naka Euclidean matrix

norm�� H jj � jjF e�nai par�moia me thn eukle�deia dianusmatik� n�rma jj � jj� all� den ep�getai ap�aut�� Gi� aut� to l�go� o �roc n�rma Frobenius e�nai protim�teroc�

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��

Parathrome �ti mia p�n�rma p�naka or�zetai me b�sh mia dianusmatik n�rma� L�me�ti h p�n�rma p�naka e�nai upotetagm�nh �subordinate norm� sthn dianusmatik p�n�rma epag�menh �induced norm� ap� thn dianusmatik p�n�rma� Ston Mn�n� hp�n�rma kale�tai ep�shc telestik� n�rma n�rma telest� �operator norm� ak�mafusik� n�rma pou antistoiqe� sth dianusmatik n�rma p�n�rma� Ant� thc ������� giath p�n�rma p�naka qrhsimopoie�tai ep�shc h isodnamh �kfrash

jjAjjp � supjjujj��

jjAujjp � A � Mm�n�K� kai u � Kn � �� �

H n�rma Frobenius �qei merik�c qr simec idi�thtec kai upolog�zetai pol ekola�Epeid �mwc den e�nai upotetagm�nh se dianusmatik n�rma� h qr sh thc se k�poiecefarmog�c mpore� na odhg sei se anakrib apotel�smata�

Se arket� bibl�a grammik c �lgebrac arijmhtik c an�lushc� bl�poume suqn��na xeqwrist� orism� thc n�rmac tetragwniko� p�naka� Stic t�sseric sunj kec touOrismo ����� prost�jentai �llec do oi opo�ec exasfal�zoun thn �isodunam�a� touMn�n me ton dianusmatik� q�ro Kn�

Orism�c �����

Kalome n�rma �tetragwniko� p naka �matrix norm� mia apeik�nish thc mor�f c jj � jj � Mn�n �� R� h opo�a� gia k�je A�B � Mn�n�K�� � � K kaiu � Kn� ikanopoie� tic ak�loujec sunj kec�B�� jjAjj � ��B�� jjAjj � � �� A � O

�B�� jj�Ajj � j�j jjAjj� � � K

�B�� jjA�Bjj � jjAjj� jjBjj�B�� jjABjj � jjAjj jjBjj�B�� H jj � jj kai h n�rma jj � jj� tou Kn pou qrhsimopoie�tai gia ton orism� thcpr�thc e�nai sumbibast�c �compatible��

jjAujj� � jjAjj jjujj� �

Parathr�seic

�� Apeikon�seic pou ikanopoion m�no tic B��B� kalontai genikeum�nec n�rmecp naka�

�� H sunj kh B� kale�tai pollaplasiastik� trigwnik� anis�thta �multiplica�tive triangular inequality��

Sth sun�qeia ja de�xoume �ti h n�rma Frobenius kaj�c kai h p�n�rma ikanopoion�lec tic sunj kec tou Orismo ������ Eidik� h p�n�rma ikanopoie� thn B� ex orismoafo gia u �� �

jjAjjp � jjAujjpjjujjp ��

Page 27: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� Q�roi me n�rma ��

jjAujjp � jjAjjp jjujjp � ����

kai gia u�� isqei am�swc h is�thta�

N�rma Frobenius

B��B� Oi sunj kec B��B� ikanopoiontai afo h jj � jjF antistoiqe� sto snhjeceswterik� gin�meno tou Mn�n�K�� �Ja mporosame ep�shc na parathr soume �ti hjj � jjF e�nai ant�stoiqh thc eukle�deiac n�rmac ston Kn� ��

B� Gia k�je A�B � Mn�n�K� pa�rnoume

jjABjjF �

nXk��

aik bkj

�F

�� nX

i��

nXj��

nX

k��

aik bkj

������

Qrhsimopoi�ntac thn anis�thta ������ pa�rnoume

jjABjjF ��� nX

i��

nXj��

�nX

k��

jaikj� �

nXk��

jbkjj� �����

��

jjABjjF ��

nXi��

nXk��

jaikj����� �� nX

j��

nXk��

jbkjj������

��

jjABjjF � jjAjjF jjBjjF �

B�� H jj � jjF e�nai ep�shc sumbibast me thn eukle�deia n�rma tou Kn� Pragmatik��gia k�je A � Mn�n�K� kai u � Kn �qoume

jjAujj� �

nXk��

aik uk

��

�� nX

i��

nX

k��

aik uk

������

��

jjAujj� ��

nXi��

�nX

k��

jaikj� �

nXk��

jukj� ����

��

jjAujj� ��

nXi��

�nX

k��

jaikj� ���� � nX

k��

jukj�����

��

jjAujj� � jjAjjF jjujj� � ����

�Ara h jj � jjF e�nai n�rma smfwna me ton Orism� ������

pn�rmaJa de�xoume pr�ta �ti h p�n�rma p�naka e�nai peperasm�nh� B�sei thc Pr�tashc ������up�rqoun stajer�c c�� c� �� t�toiec �ste gia k�je u � Kn

c� jjujj� � jjujjp � c� jjujj� � �i�

Page 28: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

E�nai faner� �ti isqei ep�shc

c� jjAujj� � jjAujjp � c� jjAujj� � �ii�

Ap� tic �i� kai �ii� pa�rnoume

jjAujjpjjujjp � c�

c�

jjAujj�jjujj�

kai� l�gw thc �������jjAujjpjjujjp � c�

c�jjAjjF �

�Ara loip�n h p�n�rma p�naka e�nai peperasm�nh��Eqontac dh de�xei �ti h B� ikanopoie�tai� ja epalhjesoume t�ra tic idi�thtec

B��B� tou Orismo ������B�� E�nai profan c ap� ton orism� thc p�n�rmac p�naka�B�� An jjAjjp��� t�te

supu���

jjAujjpjjujjp � � �� Au � � u � Kn �� A � O �

B�� Gia k�je � � K�

jj�Ajjp � supu���

jj�Aujjpjjujjp � sup

u���j�j jjAujjpjjujjp � j�j sup

u���

jjAujjpjjujjp � j�j jjAjjp �

B�� Epeid h �dianusmatik � jj � jjp e�nai n�rma ston Kn kai ikanopoie� thn trigwnik anis�thta�

jj�A � B� ujjp � jjAu � Bujjp � jjAujjp � jjBujjp �Ap� thn ������ �qoume

jj�A � B� ujjp � jjAjjp jjujjp � jjBjjp jjujjpkai� gia u �� ��

jj�A � B� ujjpjjujjp � jjAjjp � jjBjjp �� jjA � B�jjp � jjAjjp � jjBjjp �

B�� L�gw thc �������

jj�AB� ujjp � jjA �Bu�jjp � jjAjjp jjBujjp � jjAjjp jjBjjp jjujjpkai gia u �� �

jj�AB� ujjpjjujjp � jjAjjp jjBujjp �� jjABjjp � jjAjjp jjBujjp �

Gia p���� kai �qoume tic pio sunhjism�nec p�n�rmec p�naka� Gia thn �� kai thn�n�rma en�c p�naka A � Mm�n mporome na de�xoume �ti

jjAjj� � max��j�n

mXi��

jaijj ����

Page 29: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� Q�roi me n�rma �

jjAjj� � max��i�m

nXj��

jaijj � ���

H ��n�rma e�nai to m�gisto kat� st lh �jroisma twn ap�lutwn tim�n twn stoiqe�wntou p�naka kai gi�aut� kale�tai n�rma sthl�n �column norm� kai sumbol�zetai mejj � jjc� H �n�rma e�nai to m�gisto kat� gramm �jroisma twn ap�lutwn tim�n twnstoiqe�wn tou p�naka kai gi�aut� kale�tai n�rma gramm�n �row norm� kai sumbol��zetai me jj � jjr� E�nai idia�tera qr simh sthn mel�th thc sgklishc apeiroginom�nwnkaj�c kai dun�mewn stoqastik�n pinkwn� Upenjum�zoume �ti �nac pragmatik�cn � n p�nakac A e�nai stoqastik�c an e�nai mh arnhtik�c �aij � � i� j� kai an

nXj��

aij � � i �

Parathrome �ti t�te jjAjj����

Gia thn ��n�rma perioriz�maste se tetragwnikoc p�nakec A � Mn�n� H �kfrashpou pa�rnoume apaite� gn�seic thc jewr�ac idiotim�n me thn opo�a ja asqolhjomesto ep�meno kef�laio� Gia q�rh thc autonom�ac thc paragr�fou paraj�toume ed� tosqetik� apot�lesma kai k�poiouc sntomouc orismoc� Kalome f�sma �spectrum�to snolo twn idiotim�n tou A kai to sumbol�zoume me ��A�� To m�gisto m�tro twnidiotim�n tou A to kalome fasmatik� akt na �spectral radius� kai to sumbol�zoumeme r��A�

r��A� � max����A�

j�j �

Apodeiknetai �ti gia thn ��n�rma isqei

jjAjj� �qr��A�A� � ����

H ��n�rma p�naka kale�tai ep�shc fasmatik� n�rma �spectral norm�� H n�rma aut �qei k�poiec qr simec idi�thtec pou den tic �qoun h �� kai �n�rma� Parousi�zei�mwc to sobar� meion�kthma �ti den mpore� na upologisje� apeuje�ac ap� ta stoiqe�atou A� afo apaite� ton upologism� twn idiotim�n tou�

Ja de�xoume t�ra �ti h ��n�rma en�c tetragwniko p�naka A � Mn�n d�netai ap�thn ������� Wc gnwst�� gia th dianusmatik ��n�rma �qoume

jjujj� �nXi��

juij �

�Etsi�

jjAujj� �

nXk��

aik uk

� �

�nXi��

nX

k��

aik uk

�nXi��

nXk��

jaikj jukj ��

jjAujj� �nX

k��

jukjnXi��

jaikj �� jjAujj� � jjujj�nXi��

jaikj �

Page 30: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

An t�ra

c � max��k�n

nXi��

jaikj � �i�

t�te

jjAujj� � c jjujj� �� jjAujj�jjujj� � c �

Arke� t�ra na brome u � Kn t�toio �ste na isqei h is�thta� An j e�nai o de�kthcst lhc ston opo�o sumba�nei to m�gisto sthn �i� kai ej e�nai to j�ost� stoiqe�o thcsun jouc b�shc tou Kn� t�te gia u�ej �qoume jjujj� kai

jjAujj� �nXi��

nX

k��

aik uk

�nXi��

jaijj � c �

�Ara loip�n

jjAjj� � max��j�n

mXi��

jaijj �

H ap�deixh thc ������ e�nai par�moia kai af netai san �skhsh�

Pr�tash �����

�Estw oi n�rmec jj � jj� tou Rn kai jj � jj� tou Rm kai o p�nakac A � Mm�n�R��Isqei t�te

jjAujj� � jjAjj��� jjujj� � ����

�pou h n�rma jj � jj��� or�zetai ap� thn

jjAjj��� � supu ���

jjAujj�jjujj� � ����

L�me �ti h n�rma jj � jj��� e�nai upotetagm�nh stic epag�menh ap� tic dianusmatik�c n�rmec jj � jj� kai jj � jj��

H n�rma Frobenius kaj�c kai oi p�n�rmec �eidik� gia p������ ikanopoion di��forec anis�thtec pou qrhsimopoiontai suqn� sthn arijmhtik grammik �lgebra� Hap�deixh twn anisot twn pou akoloujon e�nai apl kai af netai wc �skhsh

jjAjj� � jjAjjF � pn jjAjj� � ����

maxi�j

jaijj � jjAjj� �pmn max

i�jjaijj � ����

�pnjjAjj� � jjAjj� �

pm jjAjj� � �� �

�pmjjAjj� � jjAjj� �

pn jjAjj� � ����

Page 31: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

���� Q�roi me n�rma ��

Par�deigmaGia ton p�naka

A �

��� � � �

� � �� �� �

��� �

br�skoume tic n�rmec

jjAjj� � max f � � � g �

jjAjj� � max f � � � g � �

jjAjjF �p�� � �� � � � � � �� � �� � �� � �� � �� �

p��

Page 32: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

Pr�tash �����

Gia k�je n�rma ston Mn�n isq�oun ta ex�c��a� jjIjj � ���b� An o A � Mn�n enai antistryimoc� t�te

jjA��jj � jjIjjjjAjj �

�g� Gia k�je pnaka A � Mn�n kai k � N isq�ei

jjAkjj � jjAjjk �

Ap�deixh�a� jjIjj � jjI Ijj � jjIjj jjIjj �� jjIjj � ���b� Epeid� AA���I� qoume

jjAA��jj � jjIjj �� jjAjj jjA��jj � jjIjj �� jjA��jj � jjIjjjjAjj �

�Epeid� o A enai antistryimoc� A �� O� Ara jjAjj � ����g�

jjAkjj � jjAk�� Ajj � jjAk��jj jjAjj � jjAk�� Ajj jjAjj � jjAk��jj jjAjj� �Suneqzontac brskoume th zhto�menh sqsh� �

Enai e�kolo na do�me �ti gia k�je p�n�rma pnaka isq�ei

jjIjjp � � �

Etsi� an o A � Mn�n enai antistryimoc� parnoume

jjAjjp jjA��jjp � � �

Orism�c �����

Estw mia p�n�rma stonMn�n� Kalo�me de�kth kat�stashc �condition num�ber� tou n � n pnaka A ton arijm�

� � jjAjjp jjA��jjp � ����

O dekthc kat�stashc enai dekthc thc eust�jeiac en�c grammiko� n�n sust�ma�toc kai prosdiorzei thn axiopista thc arijmhtik�c tou eplushc �p�q� me th mjodoGauss�� An o � enai mikr�c� lme �ti to s�sthma enai kalc kat�stashc �well�conditioned system�� An o � enai pol� megal�teroc tou �� t�te to s�sthma jewretaiastajc � kakc kat�stashc� �ill�conditioned or badly conditioned system��

�T�so sthn agglik� �so kai sthn ellhnik� bibliograf�a� bl�poume suqn� ton prosdiorism� kal�c� kak�c kat�stashc na apod�detai ston ant�stoiqo p�naka kai �qi sto s�sthma� to opo�o den e�naiakrib�c�

Page 33: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

Ston q�ro C�R�a� b� orzontai �metax� �llwn� oi ex�c shmantikc n�rmec�

jjf jj� �Z b

ajf�tj dt � �����

jjf jj� �

�Z b

ajf�tj� dt

����� ����

jjf jj� � maxt��a�b�

jf�tj � ����

Enai faner� �ti h jj�jj� prok�ptei ap� to s�nhjec eswterik� gin�meno ston C�R�a� b��H jj � jj� lgetai epshc n�rma meg�stou �max norm��

Par�deigmaGia thn f�x�cosx sto di�sthma ��� ��� brskoume tic n�rmec

jjf jj� �Z �

�j cos tj dt � �

Z ���

�cos t dt � �

jjf jj� ��Z �

�j cos tj� dt

�����

r�

�jjf jj� � max

t������j cos tj � �

Er�thsh� ti ekfr�zoun gewmetrik� oi jj � jj� kai jj � jj��Oi di�forec n�rmec pou orsame mqri stigm�c sunoyzontai ston Pnaka ����

Me th bo�jeia thc n�rmac� eis�getai h nnoia thc ap�stashc d metax� d�o stoi�qewn en�c q�rou me n�rma�

Orism�c �����

An �V� jj � jj nac q�roc me n�rma kai u� v � V � t�te o arijm�c

d�u� v � jj u � vjj ����

onom�zetai ap�stash �distance� twn u kai v�

An h n�rma prok�ptei ap� eswterik� gin�meno� apodeikn�etai e�kola �ti

d�u� v �hjjujj� � �Re hu� vi � jjvjj�

i���� ����

Par�deigmaH ap�stash metax� twn u� v � R� enai

d�u� v � jj u � vjj �q�u� � v�� � �u� � v�� � �u� � v�� �

Aut� enai fusik� h gnwst� ap� thn Analutik� Gewmetra ap�stash d�o shmewnston R�� Epeid� aut� antistoiqe sth d�o�n�rma� sumbolzetai epshc me d��

Page 34: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

P�nakac ��� N�rmec se di�forouc grammiko�c q�rouc�

�� Kn

p�norm jjujjp �

�nXi��

juijp���p

� � � p �

jjujj� �nXi��

juij

Euclidean jjujj� � hu � ui��� �

�nXi��

juij�����

jjujj� � max��i�n

juij

�� Mm�n�K

Frobenius jjAjjF � hA � Ai��� �qtr�A�A �

�� mXi��

nXj��

jaijj������

jjAjj �mXi��

nXj��

jaijj

p�norm jjAjjp � supu���

jjAujjpjjujjp � � � p �

jjAjj� � max��j�n

mXi��

jaijj

jjAjj� �qr��A�A � A � Mn�n

jjAjj� � max��i�m

nXj��

jaijj

�� C�K�a� b�

jjf jj� �Z b

ajf�tj dt

jjf jj� � hf � fi��� �

�Z b

ajf�tj� dt

����

jjf jj� � maxt��a�b�

jf�tj

Shme�wsh� Ta eswterik� gin�mena e�nai ta sun�jh stouc ant�stoiqouc qrouc�

Page 35: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

Sth genik�terh perptwsh tou q�rou Kn� qoume

d� � jj u � vjj� �nXi

jui � vij ����

d� � jj u � vjj� �

vuut nXi

jui � vij� ����

d� � jj u � vjj� � max��i�n

jui � vij ����

Par�deigma �Gia ta dian�smata u���� �� � kai v���� ���� tou R�� qoume�

d� � jj u � vjj� � jj��� ����jj� � � � � � � � �

d� � jj u � vjj� � jj��� ����jj� �q�� � � � ���� �

p��

d� � jj u � vjj� � jj��� ����jj� � maxf�� �� �g � �

Par�deigma Estw �ti h u � C�R�a� b� enai h akrib�c l�sh miac sun�jouc diaforik�c exswshckai �u � C�R�a� b� h prosggis� thc me k�poia arijmhtik� mjodo� gia par�deigmatwn peperasmnwn stoiqewn� H ap�stash d� twn u kai �u�

d� � jj u � �ujj� �Z b

aj u � �uj dt � ����

enai mtro tou arijmhtiko� sf�lmatoc�

Pr�tash �����

An d h ap�stash s�na q�ro me n�rma �V� jj � jj� t�te gia u� v� w � V isq�ounoi ex�c idi�thtec��G�� d�u� v � � �jetik�thta��G�� d�u� u � ��G�� d�u� v � d�v� u �summetra��G�� d�u� v � d�u�w � d�v�w �trigwnik� anis�thta�

Ap�deixhOi idi�thtec G��G� prok�ptoun e�kola ap� ton Orism� ����� thc n�rmac jj � jj� Giapar�deigma� gia thn G� qoume

d�u� v � jju � vjj � jj��� �v � ujj � j � �j jjv � ujj � jjv � ujj � d�v� u �

en� gia thn G� mporo�me na gr�youme�

d�u� v � jju � vjj � jj�u� w � �w � vjj � jju � wjj� jjv � wjj ��

Page 36: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

d�u� v � d�u�w � d�v�w �

Parathrseic

�� H is�thta sthn G� isq�ei an kai m�no an u�v�

d�u� v � � � u � v �

�� Genik�� mia apeik�nish d � V � V � R pou ikanopoie tic sunj�kec G��G� touOrismo� ����� enai mia metrik �metric�� Enac q�roc efodiasmnoc me miametrik� kaletai metrik�c q�roc �metric space�� Enai faner� �ti oi q�roi men�rma � eswterik� gin�meno enai metriko q�roi�

Orism�c �����

Estw �V� h � iv kai �W� h � iw q�roi me eswterik� gin�meno� Kalo�me iso�metrik emf�teush �isometric embedding� thn apeik�nish T � L�V�W � an

hT �u � T �viw � hu � viv �u� v � V � ���

Oi isometrikc emfute�seic apotelo�n mia pol� periorismnh kl�sh grammik�napeikonsewn� �pwc fanetai kai ap� tic prot�seic pou akoloujo�n�

Pr�tash ������

Estw �V� h � iv kai �W� h � iw q�roi me eswterik� gin�meno� An h T � L�V�W enai isometrik� emf�teush� t�te h T enai omal�� dhl� KerT�f�g�

Ap�deixh Estw u � V ttoio �ste T �u��� Epeid� h T enai isometrik� emf�teush�

hT �u � T �uiw � hu � uiv �� h� � �iw � hu � uiv �� hu � uiv � � �� u � � �

Ara KerT�f�g� �

Pr�tash ������

Estw �V� h � iv kai �W� h � iw q�roi me eswterik� gin�meno� AndimV�dimW�n � N kai h T � L�V�W enai isometrik� emf�teush� t�teh T enai nac grammik�c isomorfism�c metax� twn V kai W �

Ap�deixhAp� thn Pr�tash ������� sun�goume �ti KerT�f�g� S�mfwna me to Je�rhma ������h T enai amfimonos�manth� Tloc� epeid� dimV�dimW�n � N� h T enai ep �Je��rhma ������� Ara h T enai nac grammik�c isomorfism�c metax� twn V kai W�Orism�c ������� �

Page 37: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

��

Par�deigmaH T � L�R��R� pou orzetai ap� thn

T �u�� u� ��p��u�� u�� u�� u�

enai isometrik� emf�teush� Pragmatik�� an u� v � R�� qoume�

hT �u � T �vi �

�p��u�� u�� u�� u� �

�p��v�� v�� v�� v�

��

��u�v� � u�v� � u�v� � u�v� � u�v� � u�v� � hu � vi �

Orism�c �����

Estw �V� jj � jjv kai �W� jj � jjw q�roi me n�rma� Lme �ti oi V kai W enaiisometriko� �isometric� � isometrik� is�morfoi an up�rqei nac grammik�cisomorfism�c T � L�V�W � ttoioc �ste gia k�je u � V na isq�ei

jjT �ujjw � jjujjv � ����

O T onom�zetai t�te isometr�a �isometry� � isometrik�c isomorfism�c �iso�metric isomorphism� metax� twn V kai W�

Mia isometra se eukledeio q�ro kaletai orjog�nia apeik�nish �orthogonaltransformation��

H pr�tash pou akolouje prok�ptei e�kola ap� ta prohgo�mena�

Pr�tash ������

An �V� h�� �iv kai �W� h�� �iw enai q�roi me eswterik� gin�meno medimV�dimW�n � N kai h T � L�V�W enai isometrik� emf�teush� t�teh T enai nac isometrik�c isomorfism�c metax� twn V kai W �

Enai faner� �ti an oi V kai W enai isometrik� is�morfoi� t�te

jj T �u � T �v jjw � jj T �u� v jjw � jju � vjjv �u� v � V � ���

Sumperanoume �ti mia isometra metax� twn V kai W diathre� tic apost�seic�D�o isometrik� is�morfoi grammiko q�roi tautzontai kai wc proc ta metrik� toucstoiqea �topologik� dom���

Je�rhma ������

An oi �V� h� � �iv kai �W� h� � �iw enai peperasmenodi�statoi q�roi me eswterik�gin�meno� t�te oi �V� jj�jjv kai �W� jj�jjw� �pou jj�jjv�h�� �i���v kai jj�jjw�h�� �i���w �enai isometrik� is�morfoi ann dimV�dimW �

Page 38: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

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Ap�deixhEf�son oi V kai W enai isometrik� is�morfoi� enai kai grammik� is�morfoi op�tedimV�dimW �Je�rhma ������� Gia to antstrofo� parapmpoume ton anagn�sth stobiblo tou L� Smith� Linear Algebra� Springer�Verlag� New York� � ��� �

Ameso p�risma tou Jewr�matoc ������ enai to ak�loujo�

Je�rhma ������

K�je eukledeioc �ant� monadiaoc� grammik�c q�roc me di�stash n � N enaiisometrik� is�morfoc proc ton Rn �ant� Cn��

H shmasa tou Jewr�matoc ������ enai pol� meg�lh afo� epitrpei th metafor��lwn twn sumperasm�twn pou aforo�n sth metrik� dom� tou Rn �ant� Cn� se k�je�llo eukledeio �ant� monadiao� q�ro me thn dia di�stash�

Klenoume to par�n ed�fio me na apl� orism��

Orism�c ������

Estw �V� jj � jj q�roc me n�rma� Lme �ti to u � V enai monadia�o di�nusma�unit vector� an jjujj���

K�je mh mhdenik� stoiqeo u tou �V� jj � jj mpore na kanonikopoihje� brskontacto monadia�o bajmwt� tou pollapl�sio �u��u me � � R kai � ���

�u �u

jjujj �

To �u kaletai kanonikopoihmno �normalized�� en� h diadikasa e�res�c tou kaletaikanonikopo�hsh�

Par�deigma �Ta stoiqea thc sun�jouc b�shc tou Rn enai monadiaa� Sto sumprasma aut�katal�goume qrhsimopoi�ntac opoiad�pote ap� tic jj � jj�� jj � jj� kai jj � jj��Par�deigma Gia to u���� �� ���� � R� qoume jjujj�p� kai tsi to u kanonikopoietai wc ex�c�

�u �u

jjujj �

��p��

�p�� �� � �p

��

Par�deigma �Gia th sun�jh n�rma thc f�x�sinx � C�R��� ���� qoume

jjf�xjj �� Z ���

�sin� x dx

�����

p�

Page 39: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

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kai tsi�f�x �

�p�

sinx �

Par�deigma �Ja kanonikopoi�soume ta stoiqea thc sun�jouc b�shc tou P� sto di�sthma ��� ���ergaz�menoi me th sun�jh n�rma jj � jj�� Brskoume pr�ta tic n�rmec twn p���� p��xkai p��x��

jjp�jj �� Z �

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jjp�jj �� Z �

�x� dx

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�� x�

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jjp�jj �� Z �

�x� dx

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�� x

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Etsi ta kanonikopoihmna dian�smata enai ta ex�c�

�p� �p�jjp�jj � � � �p� �

p�jjp�jj �

p� x � �p� �

p�jjp�jj �

p� x�

Par�deigma � � Polu�numa LegendreOi l�seic thc diaforik�c ex�swshc Legendre�

��� x� y�� � �x y� � n�n� � y � � � �����

kalo�ntai sunartseic Legendre t�xhc n� An n�������� � �� oi l�seic thc pio p�nwDE enai ta polu�numa Legendre pn�x pou prok�ptoun ap� ton t�po tou Rodrigue

pn�x ��

�� n�

dn

dxn�x� � �n � �����

Ta pr�ta d�o polu�numa Legendre enai ta p��x�� kai p��x�x� Ta ep�mena polu��numa mporo�n na upologisto�n me ton anadromik� t�po�

�n� � pn��x � ��n � � x pn�x � n pn���x � � �

Eqoume tsi�

p��x � �

p��x � x

p��x ��

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p��x ��

���x� � �x

p��x ��

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Page 40: Kefl aio QWROI ME ESWTERIKOeuclid.mas.ucy.ac.cy/~georgios/courses/mas191/Chapt8.pdfmo nadiao eswterik gin me no twn u kai v H nnoia to u eswterik o gino m no u Orismc a Enac grammikc

Gia ta polu�numa Legendre isq�ei

Z �

��p�n�x dx �

�n � �� �����

op�te� orzontac th sun�jh n�rma sto ���� ��� parnoume

jjpn�xjj �s

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Mporo�me t�ra e�kola na kanonikopoi�soume ta pr�ta polu�numa Legendre�

�p��x �

p�

�� �p��x �

s�

�x � �p��x �

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