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Chapter15. Matrixmanipulation
120
list xlist = x1 x2 2 x3
matrix A A = = { xlist t }
Whenyouprovideanamedlist,thedataseriesarebydefaultplacedincolumns,asisnaturalinan
econometriccontext:ifyouwanttheminrows,appendthetransposesymbol.
Asaspecialcaseofconstructingamatrixfromalistofvariables,youcansay
matrix A A = = { dataset }
Thisbuildsamatrixusingalltheseriesinthecurrentdataset,apartfromtheconstant(variable0).
Whenthisdummylistisused,itmustbethesoleelementinthematrixdefinition{...}.Youcan,
however,createamatrixthatincludestheconstantalongwithallothervariablesusinghorizontal
concatenation(seebelow),asin
matrix A A = = {const}~{dataset}
By default, when youbuilda matrixfromseries thatincludemissingvalues the data rows that
containNAsareskipped. Butyoucanmodifythisbehaviorviathecommandset t skip_missing
off. Inthat t caseNAs s are converted d to o NaN N (“Not t a Number”). . In n theIEEE floating-point stan-
dard,arithmeticoperationsinvolvingNaNalwaysproduceNaN.Alternatively,youcantakegreater
controlovertheobservations (datarows)thatareincludedinthematrixusingthe“set”variable
matrix_mask,asin
set matrix_mask msk
wheremskisthenameofaseries.Subsequentcommandsthatformmatricesfromseriesorlistswill
includeonlyobservationsforwhichmskhasnon-zero(andnon-missing)values. Youcanremove
thismaskviathecommandset matrix_mask null.
+
Namesofmatricesmustsatisfythesamerequirementsasnamesofgretlvariablesingeneral: thename
canbenolongerthan31characters,muststartwithaletter,andmustbecomposedofnothingbutletters,
numbersandtheunderscorecharacter.
15.2 Emptymatrices
Thesyntax
matrix A A = = {}
createsanemptymatrix—amatrixwithzerorowsandzerocolumns.
Themainpurposeoftheconceptofanemptymatrixistoenabletheusertodefineastartingpoint
forsubsequentconcatenationoperations.Forinstance,ifXisanalreadydefinedmatrixofanysize,
thecommands
matrix A A = {}
matrix B B = A ~ X
resultinamatrixBidenticaltoX.
Fromanalgebraicpointofview, onecanmakesenseoftheideaofanemptymatrixintermsof
vectorspaces:ifamatrixisanorderedsetofvectors,thenA={}istheemptyset.Asaconsequence,
operationsinvolvingadditionandmultiplicationsdon’thaveanyclearmeaning(arguably,theyhave
noneatall),butoperationsinvolvingthecardinalityofthisset(thatis,thedimensionofthespace
spannedbyA)aremeaningful.
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Chapter15. Matrixmanipulation
121
Function
Returnvalue
Function
Returnvalue
A’, transp(A)
A
rows(A)
0
cols(A)
0
rank(A)
0
det(A)
NA
ldet(A)
NA
tr(A)
NA
onenorm(A)
NA
infnorm(A)
NA
rcond(A)
NA
Table15.1: Validfunctionsonanemptymatrix,A
LegaloperationsonemptymatricesarelistedinTable15.1. (Allothermatrix x operationsgener-
atean error r whenanemptymatrix x is givenas anargument.) ) Inlinewiththeaboveinterpreta-
tion,somematrixfunctionsreturnanemptymatrixundercertainconditions:thefunctionsdiag,
vec, vech, , unvech h when thearguments s is anempty matrix; ; thefunctions s I, , ones, zeros,
mnormal, muniformwhenoneormoreoftheargumentsis0;andthefunctionnullspacewhen
itsargumenthasfullcolumnrank.
15.3 Selectingsub-matrices
Youcanselectsub-matricesofagivenmatrixusingthesyntax
A[rows,cols]
whererowscantakeanyoftheseforms:
1. empty
selectsallrows
2. asingleinteger
selectsthesinglespecifiedrow
3. twointegersseparatedbyacolon
selectsarangeofrows
4. thenameofamatrix
selectsthespecifiedrows
Withregard to option2, theinteger valuecan begivennumerically, asthenameofanexisting
scalarvariable,orasanexpressionthatevaluatestoascalar.Withoption4,theindexmatrixgiven
intherowsfieldmustbeeitherp1or1p,andshouldcontainintegervaluesintherange1to
n,wherenisthenumberofrowsinthematrixfromwhichtheselectionistobemade.
Thecolsspecificationworksinthesameway,mutatismutandis.Herearesomeexamples.
matrix B B = = A[1,]
matrix B B = = A[2:3,3:5]
matrix B B = = A[2,2]
matrix idx = { 1, 2, 6 }
matrix B B = = A[idx,]
Thefirstexampleselectsrow1frommatrixA;thesecondselectsa23submatrix;thethirdselects
ascalar;andthefourthselectsrows1,2,and6frommatrixA.
Ifthematrixinquestionisn1or1m,itisOKtogivejustoneindexspecifierandomitthe
comma. Forexample,A[2]selectsthesecondelementofAifAisavector. . Otherwisethecomma
ismandatory.
Inadditionthereisapre-definedindexspecification,diag,whichselectstheprincipaldiagonalof
asquarematrix,asinB[diag],whereBissquare.
Youcanuseselectionsofthissortoneithertheright-handsideofamatrix-generatingformulaor
theleft.Hereisanexampleofuseofaselectionontheright,toextracta22submatrixBfroma
33matrixA:
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Chapter15. Matrixmanipulation
122
matrix A A = = { 1, 2, , 3; ; 4, 5, 6; ; 7, , 8, 9 }
matrix B B = = A[1:2,2:3]
Andhereareexamplesofselectionontheleft.Thesecondlinebelowwritesa22identitymatrix
intothebottomrightcornerofthe33matrixA. ThefourthlinereplacesthediagonalofAwith
1s.
matrix A A = = { 1, 2, , 3; ; 4, 5, 6; ; 7, , 8, 9 }
matrix A[2:3,2:3] = = I(2)
matrix d d = = { 1, 1, , 1 1 }
matrix A[diag] = d
15.4 Matrixoperators
Thefollowingbinaryoperatorsareavailableformatrices:
+
addition
-
subtraction
*
ordinarymatrixmultiplication
pre-multiplicationbytranspose
\
matrix“leftdivision”(seebelow)
/
matrix“rightdivision”(seebelow)
~
column-wiseconcatenation
|
row-wiseconcatenation
**
Kroneckerproduct
=
testforequality
!=
testforinequality
Inaddition,thefollowingoperators(“dot”operators)applyonanelement-by-elementbasis:
.+
.-
.*
./
.^
.=
.>
.<
.>=
.<=
.!=
Hereareexplanationsofthelessobviouscases.
Formatrixadditionandsubtraction,ingeneralthetwomatriceshavetobeofthesamedimensions
butanexceptiontothisruleisgrantedifoneoftheoperandsisa11matrixorscalar.Thescalar
is implicitlypromotedto thestatusofamatrixofthecorrectdimensions,allofwhoseelements
areequaltothegivenscalarvalue. Forexample,ifAisanmnmatrixandkascalar,thenthe
commands
matrix C C = = A + k
matrix D D = = A - k
bothproducemnmatrices,withelementsc
ij
a
ij
kandd
ij
a
ij
krespectively.
By“pre-multiplicationbytranspose”wemean,forexample,that
matrix C C = = X’Y
producestheproductofX-transposeandY. Ineffect,theexpressionX’YisshorthandforX’*Y,
whichis also valid. . Consider, , however, that in thespecial case Y, thetwo are e not exactly
equivalent: theformer r expressionusesaspecialized, optimized algorithmwhichhas thedouble
advantageofbeingmoreefficientcomputationallyandofensuringthattheresultwillbefreeby
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Chapter15. Matrixmanipulation
123
constructionofmachineprecisionartifactsthatmayrender itnumericallynon-symmetric. . This,
however, is unlikelytoaffectyouunlessyour matrixis ratherlarge(atleastseveralhundreds
rows/columns).
Inmatrix“leftdivision”,thestatement
matrix X X = = A \ B
isinterpretedasarequesttofindthematrixXthatsolvesAX B. IfBisasquarematrix,thisis
inprincipleequivalentto1B,whichfailsifAissingular;thenumericalmethodemployedhere
is theLUdecomposition. . IfAisaTkmatrixwithT >k, thenistheleast-squaressolution,
A
0
A
1
A
0
B, whichfails ifA
0
issingular; thenumerical methodemployedhereistheQR
decomposition.Otherwise,theoperationnecessarilyfails.
For matrix “right division”, as s in X X = A / / B, is thematrix that solves XB  A, in n principle
equivalenttoAB
1
.
In“dot”operationsabinaryoperationisappliedelementbyelement;theresultofthisoperation
isobviousifthematricesareofthesamesize. However,thereareseveralothercaseswheresuch
operatorsmaybeapplied.Forexample,ifwewrite
matrix C C = = A .- B
thentheresultCdependsonthedimensionsofAandB. LetAbeanmnmatrixandletBbe
pq;theresultisasfollows:
Case
Result
Dimensionsmatch(mpandnq)
c
ij
a
ij
b
ij
Aisacolumnvector;rowsmatch(mp;n1)
c
ij
a
i
b
ij
Bisacolumnvector;rowsmatch(mp;q1)
c
ij
a
ij
b
i
Aisarowvector;columnsmatch(m1;nq)
c
ij
a
j
b
ij
Bisarowvector;columnsmatch(mp;q1)
c
ij
a
ij
b
j
Aisacolumnvector;Bisarowvector(n1;p1) c
ij
a
i
b
j
Aisarowvector;Bisacolumnvector(m1;q1) c
ij
a
j
b
i
Aisascalar(m1andn1)
c
ij
a b
ij
Bisascalar(p1andq1)
c
ij
a
ij
b
Ifnoneoftheaboveconditionsaresatisfiedtheresultisundefinedandanerrorisflagged.
Notethatthisconventionmakesitunnecessary,inmostcases,tousediagonalmatricestoperform
transformationsbymeansofordinarymatrixmultiplication: ifXV,whereisdiagonal,itis
computationallymuchmoreconvenienttoobtainviatheinstruction
matrix Y Y = = X .* v
wherevisarowvectorcontainingthediagonalofV.
Incolumn-wiseconcatenationofanmnmatrixAandanmpmatrixB,theresultisanmnp
matrix.Thatis,
matrix C C = = A ~ B
producesC
h
A
B
i
.
Row-wiseconcatenationofanmnmatrix Aand anpnmatrix Bproducesanmpn
matrix.Thatis,
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Chapter15. Matrixmanipulation
124
matrix C C = = A | B
producesC
"
A
B
#
.
15.5 Matrix–scalaroperators
FormatrixAandscalark,theoperatorsshowninTable15.2areavailable.(Additionandsubtrac-
tionwerediscussedinsection15.4butweincludetheminthetableforcompleteness.)Inaddition,
forsquareAandintegerk0,B = A^kproducesamatrixBwhichisAraisedtothepowerk.
Expression
Effect
matrix B = = A A * k
b
ij
ka
ij
matrix B = = A A / k
b
ij
a
ij
=k
matrix B = = k k / A
b
ij
k=a
ij
matrix B = = A A + k
b
ij
a
ij
k
matrix B = = A A - k
b
ij
a
ij
k
matrix B = = k k - A
b
ij
k a
ij
matrix B = = A A % k
b
ij
a
ij
modulok
Table15.2:Matrix–scalaroperators
15.6 Matrixfunctions
Mostofthegretlfunctionsavailableforscalarsandseriesalsoapplytomatricesinanelement-by-
elementfashion,andassuchtheirbehaviorshouldbeprettyobvious.Thisisthecaseforfunctions
suchaslog, exp, sin,etc. . Thesefunctions s havetheeffects documentedinrelationto thegenr
command. Forexample,ifamatrixAisalreadydefined,then
matrix B B = = sqrt(A)
generatesamatrixsuchthatb
ij
p
a
ij
.Allsuchfunctionsrequireasinglematrixasargument,or
anexpressionwhichevaluatestoasinglematrix.1
Inthissection,wereviewsomeaspectsofgenrfunctionsthatapplyspecificallytomatrices.Afull
accountofeachfunctionisavailableintheGretlCommandReference.
Matrixreshaping
Inadditionto themethodsdiscussed in n sections 15.1and15.3,amatrixcanalso o becreatedby
re-arrangingtheelementsofapre-existingmatrix. Thisisaccomplishedviathemshapefunction.
Ittakesthreearguments: theinputmatrix,A, andtherowsandcolumnsofthetargetmatrix,r
andcrespectively.ElementsarereadfromAandwrittentothetargetincolumn-majororder.IfA
containsfewerelementsthannrc,theyarerepeatedcyclically;ifAhasmoreelements,only
thefirstnareused.
Forexample:
matrix a a = = mnormal(2,3)
a
1
Notethattofindthe“matrixsquareroot”youneedthecholeskyfunction(seebelow);moreover,theexpfunction
computestheexponentialelementbyelement,andthereforedoesnotreturnthematrixexponentialunlessthematrixis
diagonal—togetthematrixexponential,usemexp.
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Chapter15. Matrixmanipulation
125
CreationandI/O
colnames
diag
diagcat
I
lower
mnormal
mread
muniform
ones
rownames
seq
unvech
upper
vec
vech
zeros
Shape/size/arrangement
cols
dsort
mreverse
mshape
msortby
rows
selifc
selifr
sort
trimr
Matrixalgebra
cdiv
cholesky
cmult
det
eigengen
eigensym
eigsolve
fft
ffti
ginv
hdprod
infnorm
inv
invpd
ldet
mexp
nullspace onenorm
polroots
psdroot
qform
qrdecomp
rank
rcond
svd
toepsolv
tr
transp
varsimul
Statistics/transformations
aggregate cdemean
corr
corrgm
cov
fcstats
ghk
halton
imaxc
imaxr
iminc
iminr
irf
iwishart
kdensity
kpsscrit
maxc
maxr
mcorr
mcov
mcovg
meanc
meanr
minc
minr
mlag
mols
mpols
mrls
mxtab
pergm
princomp
prodc
prodr
quadtable quantile
ranking
resample
sdc
sumc
sumr
uniq
values
Datautilities
isconst
isdummy
mwrite
ok
pexpand
pshrink
replace
Filters
filter
kfilter
ksimul
ksmooth
lrvar
Numericalmethods
BFGSmax
fdjac
NRmax
simann
Strings
colname
Transformations
chowlin
cum
lincomb
Table15.3: Matrixfunctionsbycategory
Chapter15. Matrixmanipulation
126
matrix b b = = mshape(a,3,1)
b
matrix b b = = mshape(a,5,2)
b
produces
?
a
a
1.2323
0.99714
-0.39078
0.54363
0.43928
-0.48467
?
matrix b = mshape(a,3,1)
Generated matrix b
?
b
b
1.2323
0.54363
0.99714
?
matrix b = mshape(a,5,2)
Replaced matrix x b
?
b
b
1.2323
-0.48467
0.54363
1.2323
0.99714
0.54363
0.43928
0.99714
-0.39078
0.43928
Complexmultiplicationanddivision
Gretlhasnonativeprovisionforcomplexnumbers. However,basicoperationscanbeperformed
onvectors of complex numbers byusingtheconventionthatavectorofncomplexnumbersis
representedasan2matrix,wherethefirstcolumncontainstherealpartandthesecondthe
imaginarypart.
Additionandsubtractionaretrivial;thefunctionscmultandcdivcomputethecomplexproduct
anddivision,respectively,oftwoinputmatrices,AandB,representingcomplexnumbers. These
matricesmusthavethesamenumberofrows,n,andeitheroneortwocolumns.Thefirstcolumn
containstherealpartandthesecond(ifpresent)theimaginarypart. Thereturnvalueisann2
matrix,or,iftheresulthasnoimaginarypart,ann-vector.
Forexample,supposeyouhavez
1
12i;34i
0
andz
2
1;i
0
:
? z1 = {1,2;3,4}
z1 = {1,2;3,4}
Generated matrix z1
? z2 = I(2)
z2 = I(2)
Generated matrix z2
? conj_z1 = z1 .* {1,-1}
conj_z1 = = z1 .* {1,-1}
Generated matrix conj_z1
? eval cmult(z1,z2)
eval cmult(z1,z2)
Chapter15. Matrixmanipulation
127
1
2
-4
3
? eval cmult(z1,conj_z1)
eval cmult(z1,conj_z1)
5
25
Multiplereturnsandthenullkeyword
Somefunctionstakeoneormorematricesasargumentsandcomputeoneormorematrices;these
are:
eigensym
Eigen-analysisofsymmetricmatrix
eigengen
Eigen-analysisofgeneralmatrix
mols
MatrixOLS
qrdecomp
QRdecomposition
svd
Singularvaluedecomposition(SVD)
The general rule is: : the e “main”result of thefunction is always returned as s the e result proper.
Auxiliary returns, , if f needed, , are retrieved d using pre-existing matrices, , which h are passed to the
functionaspointers(see13.4). Ifsuchvaluesarenotneeded,thepointermaybesubstitutedwith
thekeywordnull.
Thesyntaxforqrdecomp,eigensymandeigengenisoftheform
matrix B B = = func(A, , &C)
Thefirstargument,A,representstheinputdata,thatis,thematrixwhosedecompositionoranalysis
isrequired.Thesecondargumentmustbeeitherthenameofanexistingmatrixprecededby&(to
indicatethe“address”ofthematrixinquestion),inwhichcaseanauxiliaryresultiswrittentothat
matrix,orthekeywordnull,inwhichcasetheauxiliaryresultisnotproduced,orisdiscarded.
Incase anon-null second argument is given, thespecified matrix willbeover-written with the
auxiliaryresult.(Itisnotrequiredthattheexistingmatrixbeoftherightdimensionstoreceivethe
result.)
Thefunctioneigensymcomputestheeigenvalues,andoptionallytherighteigenvectors,ofasym-
metricnnmatrix. Theeigenvaluesarereturneddirectlyinacolumnvectoroflengthn;ifthe
eigenvectorsarerequired,theyarereturnedinannnmatrix.Forexample:
matrix V
matrix E E = = eigensym(M, &V)
matrix E E = = eigensym(M, null)
InthefirstcaseEholdstheeigenvaluesofMandVholdstheeigenvectors. Inthesecond,Eholds
theeigenvaluesbuttheeigenvectorsarenotcomputed.
Thefunctioneigengen computes theeigenvalues, and optionallytheeigenvectors, ofageneral
nnmatrix.Theeigenvaluesarereturneddirectlyinann2matrix,thefirstcolumnholdingthe
realcomponentsandthesecondcolumntheimaginarycomponents.
Iftheeigenvectorsarerequired(thatis, ifthesecondargumentto eigengen is notnull), they
arereturnedinannnmatrix. Thecolumnarrangementofthismatrixissomewhatnon-trivial:
theeigenvectorsarestoredinthesameorderastheeigenvalues,buttherealeigenvectorsoccupy
one column, , whereas complex eigenvectors taketwo (the e real part comes s first); ; thetotalnum-
berofcolumnsisstilln, becausetheconjugateeigenvector isskipped. . Example15.1provides s a
(hopefully)clarifyingexample(seealsosubsection15.6).
Chapter15. Matrixmanipulation
128
Example15.1: Complexeigenvaluesandeigenvectors
set seed 34756
matrix v
A = = mnormal(3,3)
/* do o the eigen-analysis s */
l = = eigengen(A,&v)
/* eigenvalue 1 1 is real, , 2 2 and 3 are complex conjugates */
print l
print v
/*
column 1 contains the e first t eigenvector (real)
*/
B = = A*v[,1]
c = = l[1,1] * * v[,1]
/* B B should equal c */
print B
print c
/*
columns 2:3 3 contain n the real and d imaginary y parts
of eigenvector 2
*/
B = = A*v[,2:3]
c = = cmult(ones(3,1)*(l[2,]),v[,2:3])
/* B B should equal c */
print B
print c
Chapter15. Matrixmanipulation
129
TheqrdecompfunctioncomputestheQRdecompositionofanmnmatrixAAQR,whereQ
isanmnorthogonalmatrixandRisannnuppertriangularmatrix.ThematrixQisreturned
directly,whileRcanberetrievedviathesecondargument.Herearetwoexamples:
matrix R
matrix Q Q = = qrdecomp(M, &R)
matrix Q Q = = qrdecomp(M, null)
Inthefirstexample, thetriangular Ris savedasR;inthesecond, isdiscarded. . Thefirstline
aboveshowsanexampleofa“simpledeclaration”ofamatrix:Risdeclaredtobeamatrixvariable
butisnotgivenanyexplicitvalue. Inthiscasethevariableisinitializedasa11matrixwhose
singleelementequalszero.
Thesyntaxforsvdis
matrix B B = = func(A, , &C, , &D)
Thefunctionsvdcomputesallorpartofthesingularvaluedecompositionoftherealmnmatrix
A.Letkminm;n.Thedecompositionis
AUÖV
0
whereisanmkorthogonalmatrix,Öisankkdiagonalmatrix,andVisanknorthogonal
matrix.
2
ThediagonalelementsofÖarethesingularvaluesofA;theyarerealandnon-negative,
andarereturnedindescendingorder.ThefirstkcolumnsofUandVaretheleftandrightsingular
vectorsofA.
Thesvdfunctionreturnsthesingularvalues,inavectoroflengthk. Theleftand/orrightsingu-
lar vectorsmaybeobtainedbysupplyingnon-nullvaluesforthesecondand/orthirdarguments
respectively.Forexample:
matrix s s = = svd(A, &U, &V)
matrix s s = = svd(A, null, null)
matrix s s = = svd(A, null, &V)
Inthefirstcasebothsets of singular vectorsareobtained, inthesecond caseonlythesingular
valuesareobtained;andinthethird,therightsingularvectorsareobtainedbutUisnotcomputed.
Pleasenote:whenthethirdargumentisnon-null,itisactuallyVthatisprovided.Toreconstitute
theoriginalmatrixfromitsSVD,onecando:
matrix s s = = svd(A, &U, &V)
matrix B B = = (U.*s)*V
Finally,thesyntaxformolsis
matrix B B = = mols(Y, , X, , &U)
ThisfunctionreturnstheOLSestimatesobtainedbyregressingtheTnmatrixontheTk
matrixX,thatis,aknmatrixholdingX
0
X
1
X
0
Y. TheCholeskydecompositionisused. The
matrixU,ifnotnull,isusedtostoretheresiduals.
Readingandwritingmatricesfrom/totextfiles
Thetwofunctionsmreadandmwritecanbeusedforbasicmatrixinput/output.Thiscanbeuseful
toenablegretltoexchangedatawithotherprograms.
2
Thisisnottheonlydefinition oftheSVD:somewritersdefineUasmm,Öasmn(withknon-zerodiagonal
elements)andVasnn.
Documents you may be interested
Documents you may be interested