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ChapterV
51
Whydiscussthisnow?Whatis=
q
7
3
?Hardtosayexactly,butitisdenitely
notarationalnumber.Normsofvectorswillfeatureprominentlyinallourdiscussions
aboutorthogonalvectors,sowenowhavetorecognizetheneedtoworkwithsquare
rootsproperly. WehavetwostrategiesinSage.
ThenumbersystemQQbar,alsoknownasthe\eldofalgebraicnumbers,"isa
trulyamazingfeatureofSage. Itcontains s therationalnumbers,plusevery y rootof
everypolynomialwithcoecientsthatarerationalnumbers.Forexample,noticethat
aboveisonesolutiontothepolynomialequation3x
2
7=0andthusisanumber
in QQbar, so Sage can work k with h it exactly. . These e numbers s are called\algebraic
numbers"and youcanrecognizethemsince theyprintwithaquestion mark k near
theendtoremindyouthatwhenprintedas adecimalthey areapproximationsof
numbersthatSagecarriesinternallyasexactquantities.Forexamplecanbecreated
withQQbar(sqrt(7/3))andwillprintas1.527525231651947?.Noticethatcomplex
numbersbeginwiththeintroductionoftheimaginarynumberi,whichis arootof
thepolynomialequationx
2
+1=0,sotheeldofalgebraicnumberscontainsmany
complexnumbers. ThedownsideofQQbaristhatcomputationsareslow(relatively
speaking),sothisnumbersystemismostusefulforexamplesanddemonstrations.
Theotherstrategyistoworkstrictlywithapproximatenumbers,cognizantofthe
potentialforinaccuracies.Sagehastwosuchnumbersystems:RDFandCDF,whichare
comprisedof\doubleprecision" oatingpointnumbers,rstlimitedtojustthereals,
thenexpandedtothecomplexes.Double-precisionreferstotheuseof64bitstostore
thesign,mantissaandexponentintherepresentationofarealnumber. Thisgives
53bitsofprecision. DonotconfusetheseeldswithRR R andCC,whicharesimilar
in appearance but very y dierent inimplementation. . Sage e has implementations of
severalcomputationsdesignedexclusivelyfor RDFandCDF,suchasthenorm. . And
theyare very,very fast. . But t somecomputations,likeechelonform,canbewildly
unreliablewiththeseapproximatenumbers. Wewillhavemoretosayaboutthisas
wego. Inpractice,youcanuseCDF,sinceRDFisasubsetandonlydierentinvery
limitedcases.
Insummary,QQbar isanextensionofQQwhichallows exact computations,but
canbe slowfor large examples. . RDF F andCDF are fast, , withspecialalgorithms s to
controlmuchoftheimprecisioninsome,butnotall,computations.Soweneedtobe
vigilantandskepticalwhenweworkwiththeseapproximatenumbers. Wewilluse
bothstrategies,asappropriate.
CNIPConjugates,NormsandInnerProducts
Conjugates,ofcomplexnumbersandofvectors,arestraightforward,inQQbarorin
CDF.
sage: alpha = = QQbar(2 2 + + 3*I)
sage: alpha.conjugate()
2 - - 3*I
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ChapterV
52
sage: beta = = CDF(2+3*I)
sage: beta.conjugate()
2.0 - - 3.0*I
sage: v v = vector(QQbar, [5-3*I, , 2+6*I])
sage: v.conjugate()
(5 + 3*I, 2 2 - - 6*I)
sage: w w = vector(CDF, , [5-3*I, , 2+6*I])
sage: w.conjugate()
(5.0 + + 3.0*I, , 2.0 - 6.0*I)
Theterm\innerproduct"meansslightlydierentthingstodierentpeople.For
some,itisthe\dotproduct"thatyoumayhaveseeninacalculusorphysicscourse.
Ourinnerproductcouldbecalledthe\Hermitianinnerproduct"toemphasizethe
useofvectorsoverthecomplexnumbersandconjugatingsomeoftheentries.SoSage
has a.dot_product(),.inner_product(),and.hermitian_inner_product()|
wewanttousethelastone.
Furthermore, Sage e denes s the Hermitian n inner product t by y conjugating entries
fromtherstvector,ratherthanthesecondvectorasdenedinthetext.Thisisnot
asbigaproblemasitmightseem. First,TheoremIPAC,tellsusthatwecancoun-
teractadierenceinorderbyjusttakingtheconjugate,soyoucantranslatebetween
Sageandthetextbytakingconjugatesofresults achievedfrom aHermitianinner
product. Second,wewillmostlybeinterestedinwhenaHermitianinnerproductis
zero,whichisitsownconjugate,sonoadjustmentisrequired.Third,mosttheorems
aretrueasstatedwitheitherdenition,althoughthereareexceptions,likeTheorem
IPSM.
Fromnowon,whenwementionaninner productinthecontext ofusingSage,
wewillmean.hermitian_inner_product().WewillredotherstpartofExample
CSIP.Noticethatthesyntaxisabitasymmetric.
sage: u u = vector(QQbar, [2+3*I, , 5+2*I, , -3+I])
sage: v v = vector(QQbar, [1+2*I, , -4+5*I, , 5*I])
sage: u.hermitian_inner_product(v)
3 + + 19*I
Again,noticethattheresultistheconjugateofwhatwehaveinthetext.
Normsareaseasyasconjugates.Easiermaybe.Itmightbeusefultorealizethat
SageusesentirelydistinctcodetocomputeanexactnormoverQQbarversusanap-
proximatenormoverCDF,thoughthatistotallytransparentasyouissuecommands.
HereisExampleCNSVreprised.
sage: entries s = = [3+2*I, , 1-6*I, 2+4*I, , 2+I]
sage: u u = vector(QQbar, entries)
sage: u.norm()
8.66025403784439?
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ChapterV
53
sage: u u = vector(CDF, , entries)
sage: u.norm()
8.66025403784
sage: numerical_approx(5*sqrt(3), , digits = = 30)
8.66025403784438646763723170753
We have e three e dierent numerical approximations, , the e latter r 30-digit number
being an approximationto the answer in the text. . But t there is no inconsistency
betweenthem.Therst,analgebraicnumber,isrepresentedinternallyas5awhere
aisarootofthepolynomialequationx
2
3=0,inotherwordsitis5
p
3.TheCDF
value printswithafewdigits lessthanwhatiscarriedinternally. . Noticethatour
dierentdenitionsoftheinnerproductmakenodierenceinthecomputationofa
norm.
Onewarningnowthatweareworkingwithcomplexnumbers.Itiseasyto\clob-
ber"thesymbolIusedfortheimaginarynumberi.Inotherwords,Sagewillallow
youtoassignittosomethingelse,renderingituseless.Anidentitymatrixisalikely
reassignment.Ifyourunthenextcomputecell,besuretoevaluatethecomputecell
afterwardtorestoreItoitsusualrole.
sage: alpha = = QQbar(5 5 - - 6*I)
sage: I I = identity_matrix(2)
sage: beta = = QQbar(2+5*I)
Traceback (most t recent call last):
...
TypeError: Illegal l initializer for algebraic number
sage: restore()
sage: I^2
-1
WewillnishwithavericationofTheoremIPN.Totestequalityitisbestifwe
workwithentriesfromQQbar.
sage: v v = vector(QQbar, [2-3*I, , 9+5*I, , 6+2*I, 4-7*I])
sage: v.hermitian_inner_product(v) ) == = v.norm()^2
True
OGSOrthogonalityandGram-Schmidt
Itis easyenoughtocheckapair ofvectorsfororthogonality (istheinnerproduct
zero?).Tocheckthatasetisorthogonal,wejustneedtodothisrepeatedly.Thisis
aredoofExampleAOS.
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ChapterV
54
sage: x1 1 = = vector(QQbar, , [
1+I,
1,
1-I,
I])
sage: x2 2 = = vector(QQbar, , [ [ 1+5*I,
6+5*I,
-7-I,
1-6*I])
sage: x3 3 = = vector(QQbar, , [-7+34*I, -8-23*I, -10+22*I, 30+13*I])
sage: x4 4 = = vector(QQbar, , [ [ -2-4*I,
6+I,
4+3*I,
6-I])
sage: S S = [x1, x2, x3, x4]
sage: ips s = = [S[i].hermitian_inner_product(S[j])
...
for i in range(3) ) for j j in n range(i+1,3)]
sage: all([ip p == = 0 0 for r ip p in ips])
True
Noticehowthelistcomprehensioncomputeseachpairjustonce,andneverchecks
theinnerproductofavectorwithitself.Ifwewantedtocheckthatasetisorthonor-
mal,the\normal"partislessinvolved. Wewillcheckthesetabove,eventhoughwe
canclearlyseethatthefourvectorsarenotevenclosetobeingunitvectors.Besure
toruntheabovedenitionsofSbeforerunningthenextcomputecell.
sage: ips s = = [S[i].hermitian_inner_product(S[i]) for r i i in n range(3)]
sage: all([ip p == = 1 1 for r ip p in ips])
False
ApplyingtheGram-Schmidtproceduretoasetofvectorsisthetypeofcompu-
tationthataprogram like Sageis perfectfor. . Gram-Schmidtis s implemented as a
method for matrices, where we interpret the rows s of f the matrix as the vectors s in
theoriginalset. Theresultistwomatrices,where e the rsthas rows that are the
orthogonalvectors.Thesecondmatrixhasrowsthatprovidelinearcombinationsof
theorthogonalvectorsthatequaltheoriginalvectors. Theoriginalvectorsdonot
needtoformalinearlyindependentset,andwhenthesetislinearlydependent,then
zerovectorsproducedarenotpartofthereturnedset.
Over CDF the setis automaticallyorthonormal, , and sincea dierent t algorithm
is used(tohelpcontrolthe imprecisions),theresults willlook dierentthanwhat
wouldresultfromTheoremGSP.WewillillustratewiththevectorsfromExample
GSTV.
sage: v1 1 = = vector(CDF, [ [ 1, , 1+I,
1])
sage: v2 2 = = vector(CDF, [-I,
1, 1+I])
sage: v3 3 = = vector(CDF, [ [ 0,
I,
I])
sage: A A = matrix([v1,v2,v3])
sage: G, , M M = = A.gram_schmidt()
sage: G.round(5)
[
-0.5
-0.5 - - 0.5*I
-0.5]
[ 0.30151 1 + + 0.45227*I -0.15076 6 + + 0.15076*I -0.30151 - - 0.75378*I]
[
0.6396 + + 0.2132*I
-0.2132 - - 0.6396*I
0.2132 + + 0.2132*I]
WeformedthematrixAwiththethreevectorsasrows,andofthetwooutputs
weareinterestedintherstone,whoserowsformtheorthonormalset. Weround
thenumbers to5digits,justtomaketheresulttnicelyonyourscreen. . Let’sdo
it again, , nowexactly y over QQbar. . Wewilloutput t theentries of the matix as list,
workingacrossrowsrst,soittsnicely.
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ChapterV
55
sage: v1 1 = = vector(QQbar, , [ [ 1, , 1+I,
1])
sage: v2 2 = = vector(QQbar, , [-I,
1, 1+I])
sage: v3 3 = = vector(QQbar, , [ [ 0,
I,
I])
sage: A A = matrix([v1,v2,v3])
sage: G, , M M = = A.gram_schmidt(orthonormal=True)
sage: Sequence(G.list(), , cr=True)
[
0.50000000000000000?,
0.50000000000000000? + + 0.50000000000000000?*I,
0.50000000000000000?,
-0.3015113445777636? - - 0.4522670168666454?*I,
0.1507556722888818? - - 0.1507556722888818?*I,
0.3015113445777636? + + 0.7537783614444091?*I,
-0.6396021490668313? - - 0.2132007163556105?*I,
0.2132007163556105? + + 0.6396021490668313?*I,
-0.2132007163556105? - - 0.2132007163556105?*I
]
Notice that we asked for orthonormaloutput,so therows ofG are thevectors
fw
1
;w
2
;w
3
ginExampleONTV.Exactly.WecanrestrictourselvestoQQandforego
the\normality"toobtainjusttheorthogonalsetfu
1
;u
2
;u
3
gofExampleGSTV.
sage: v1 1 = = vector(QQbar, , [ [ 1, , 1+I,
1])
sage: v2 2 = = vector(QQbar, , [-I,
1, 1+I])
sage: v3 3 = = vector(QQbar, , [ [ 0,
I,
I])
sage: A A = matrix([v1, , v2, , v3])
sage: G, , M M = = A.gram_schmidt(orthonormal=False)
sage: Sequence(G.list(), , cr=True)
[
1,
I + + 1,
1,
-0.50000000000000000? - - 0.75000000000000000?*I,
0.25000000000000000? - - 0.25000000000000000?*I,
0.50000000000000000? + + 1.2500000000000000?*I,
-0.2727272727272728? - - 0.0909090909090909?*I,
0.0909090909090909? + + 0.2727272727272728?*I,
-0.0909090909090909? - - 0.0909090909090909?*I
]
NoticethatitisanerrortoaskforanorthonormalsetoverQQsinceyoucannot
expecttotakesquarerootsofrationalsandstickwithrationals.
sage: v1 1 = = vector(QQ, [1, , 1])
sage: v2 2 = = vector(QQ, [2, , 3])
sage: A A = matrix([v1,v2])
ChapterV
56
sage: G, , M M = = A.gram_schmidt(orthonormal=True)
Traceback (most t recent call last):
...
TypeError: QR R decomposition n unable e to o compute e square e roots in n Rational Field
M:Matrices
MSMatrixSpaces
SagedenesoursetM
mn
as a\matrixspace"withthecommandMatrixSpace(R,
m, n)whereRisanumbersystemandmandnarethenumberofrowsandnumber
ofcolumns, respectively. . This s object does nothave muchfunctionality dened in
Sage.Itsmainpurposesaretoprovideaparentformatrices,andtoprovideanother
waytocreatematrices.Thetwomatrixoperationsjustdened(additionandscalar
multiplication) are implemented as you would expect. . In n the e example e below, , we
create two matrices in M
2;3
from just a list of 6 entries, , by coercing the e list into
amatrix by usingtherelevantmatrixspaceasifitwereafunction. . Thenwecan
performthebasicoperationsofmatrixaddition(DenitionMA)andmatrix scalar
multiplication(DenitionMSM).
sage: MS S = = MatrixSpace(QQ, 2, , 3)
sage: MS
Full MatrixSpace e of f 2 2 by y 3 3 dense matrices over Rational l Field
sage: A A = MS([1, , 2, , 1, 4, 5, 4])
sage: B B = MS([1/1, 1/2, 1/3, 1/4, 1/5, 1/6])
sage: A A + B
[
2 5/2 4/3]
[17/4 26/5 25/6]
sage: 60*B
[60 30 0 20]
[15 12 2 10]
sage: 5*A A - - 120*B
[-115 -50 -35]
[ -10
1
0]
Coercioncanmakesomeinterestingconveniencespossible.Noticehowthescalar
37inthefollowingexpressioniscoercedto37timesanidentitymatrixoftheproper
size.
sage: A A = matrix(QQ, [[ 0, , 2, , 4],
...
[ 6, , 0, , 8],
57
ChapterM
58
...
[10, 12, 0]])
sage: A A + 37
[37 2 2 4]
[ 6 6 37 7 8]
[10 12 2 37]
Thiscoerciononlyappliestosumswithsquarematrices.Youmighttrythisagain,
butwitharectangularmatrix,justtoseewhattheerrormessagesays.
MOMatrixOperations
Every operation n in this section is implemented in Sage. . The e only realsubtlety is
determining if certain matrices s are symmetric, , which h we will discuss below. . In
linear algebra, , theterm m \adjoint"hastwounrelatedmeanings,so youneedto be
careful when you see e this s term. . In n particular, , in n Sage e it is used d to mean some-
thingdierent. Soourversionoftheadjointisimplementedasthematrixmethod
.conjugage_transpose(). Herearesomestraightforwardexamples.
sage: A A = matrix(QQ, [[-1, , 2, , 4],
...
[ 0, , 3, , 1]])
sage: A
[-1 2 2 4]
[ 0 0 3 3 1]
sage: A.transpose()
[-1 0]
[ 2 2 3]
[ 4 4 1]
sage: A.is_symmetric()
False
sage: B B = matrix(QQ, [[ 1, , 2, , -1],
...
[ 2, , 3, , 4],
...
[-1, 4, , -6]])
sage: B.is_symmetric()
True
sage: C C = matrix(QQbar, [[ [ 2-I, , 3+4*I],
...
[5+2*I,
6]])
sage: C.conjugate()
[2 + 1*I 3 3 - - 4*I]
[5 - 2*I
6]
ChapterM
59
sage: C.conjugate_transpose()
[2 + 1*I 5 5 - - 2*I]
[3 - 4*I
6]
With these constructions, we e can n test, or r demonstrate, some of the e theorems
above.Ofcourse,thisdoesnotmakethetheoremstrue,butissatisfyingnonetheless.
ThiscanbeaneectivetechniquewhenyouarelearningnewSagecommandsornew
linearalgebra|ifyourcomputationsarenotconsistentwiththeorems,thenyour
understandingofthelinearalgebramaybe awed,oryourunderstandingofSagemay
be awed,orSage mayhave abug! ! Noteinthe e followinghowweuse comparison
(==)betweenmatricesasanimplementationofmatrixequality(DenitionME).
sage: A A = matrix(QQ, [[ 1, , -1, , 3],
...
[-3, 2, , 0]])
sage: B B = matrix(QQ, [[5, -2, , 7],
...
[1, 3, , -2]])
sage: C C = matrix(QQbar, [[2+3*I, , 1 1 - - 6*I], [3, 5+2*I]])
sage: A.transpose().transpose() == = A
True
sage: (A+B).transpose() ) == = A.transpose() ) + + B.transpose()
True
sage: (2*C).conjugate() ) == = 2*C.conjugate()
True
sage: a a = QQbar(3 + 4*I)
sage: acon = = a.conjugate()
sage: (a*C).conjugate_transpose() ) == = acon*C.conjugate_transpose()
True
Theoppositeistrue|youcanusetheoremstoconvert,orexpress,Sagecode
intoalternative,butmathematicallyequivalentforms.
Hereisthesubtlety. Withapproximatenumbers,suchasinRDFandCDF,itcan
betrickytodecideiftwonumbersareequal,orifaverysmallnumberiszeroornot.
InthesesituationsSageallowsustospecifya\tolerance"|thelargestnumberthat
canbeeectivelyconsideredzero.Considerthefollowing:
sage: A A = matrix(CDF, , [[1.0, , 0.0], [0.0, 1.0]])
sage: A
[1.0
0]
[ 0 0 1.0]
sage: A.is_symmetric()
True
ChapterM
60
sage: A[0,1] ] = 0.000000000002
sage: A
[ 1.0 0 2e-12]
[
0
1.0]
sage: A.is_symmetric()
False
sage: A[0,1] ] = 0.000000000001
sage: A
[ 1.0 0 1e-12]
[
0
1.0]
sage: A.is_symmetric()
True
Clearlythelastresultisnotcorrect.Thisisbecause0:000000000001=1:010
12
is \small enough" " to be e confused d as s equal to o the zero in the e other r corner of the
matrix. However,Sagewillletussetourownideaofwhentwonumbersareequal,
bysettingatoleranceonthedierencebetweentwonumbersthatwillallowthemto
beconsideredequal. Thedefaulttoleranceissetat1:010
12
. HereweuseSage’s
syntaxforscienticnotationtospecifythetolerance.
sage: A A = matrix(CDF, , [[1.0, , 0.0], [0.0, 1.0]])
sage: A.is_symmetric()
True
sage: A[0,1] ] = 0.000000000001
sage: A.is_symmetric()
True
sage: A.is_symmetric(tol=1.0e-13)
False
Thisisnotacourseinnumericallinearalgebra,evenifthatisafascinatingeld
ofstudy. Toconcentrateonthemainideasofintroductorylinearalgebra,whenever
possiblewewillconcentrateonnumbersystemsliketherationalnumbersoralgebraic
numberswherewecanrelyonexactresults.Ifyouareeverunsureifanumbersystem
isexactornot,justask.
sage: QQ.is_exact()
True
sage: RDF.is_exact()
False
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