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Linear Algebra and its Applications
www.elsevier.com/locate/laa
Products of positive semi-definite matrices
Jianlian Cuia, Chi-Kwong Lib,∗,Nung-Sing Szec
aDepartmentofMathematics,TsinghuaUniversity,Beijing100084,PRChina
bDepartmentofMathematics,CollegeofWilliamandMary,Williamsburg, VA 23187,USA
cDepartmentofAppliedMathematics,The HongKongPolytechnicUniversity, HongKong
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received30June2015 Accepted19September2015 Availableonlinexxxx SubmittedbyP.Semrl
DedicatedtoProfessorRajendra Bhatia
MSC:
15A23 15B48 15A60
Keywords:
Positivesemi-definitematrices Numericalrange
Itisknownthateverycomplexsquarematrixwithnonnega- tive determinant is the product of positive semi-definite matrices.Therearecharacterizationsofmatricesthatrequire two or five positive semi-definite matrices in the product.
However,thecharacterizationsofmatricesthatrequirethree or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results,we give an algorithmto determine whethera squarematrix can be expressedastheproductofkpositivesemi-definitematrices butnotfewer,fork= 1,2,3,4,5.
© 2015ElsevierInc.All rights reserved.
1. Introduction
LetMn be theset of n×ncomplex matrices. In[3], theauthor showed thatama- trixinMn with nonnegative determinantcanalways be writtenas the product offive
* Correspondingauthor.
E-mailaddresses:[email protected](J. Cui),[email protected](C.-K. Li), [email protected](N.-S. Sze).
http://dx.doi.org/10.1016/j.laa.2015.09.045 0024-3795/© 2015ElsevierInc.All rights reserved.
or fewer positivesemi-definitematrices.This is anextension tothe resultin[1]assert- ing that everymatrix in Mn with positive determinantis the product of five or fewer positive definite matrices. Analogous to the analysis in [1], the author of [3] studied those matrices which can be expressed as the product of two, three, or four positive semi-definitematrices.Inparticular,characterizationwasobtainedforthematricesthat can be expressed as theproduct oftwo semi-definite matrices; also, itwas shownthat any matricesnot of theform zI, where z is notanonnegative number, could be writ- ten as the product of four positive semi-definite matrices. Moreover, it was proved in [3, Theorem 3.3]thatifoneapplies aunitary similarityto changethesquarematrixto theformT =
T1 ∗ 0 T2
∈MnsuchthatT1isinvertibleandT2isnilpotent,andifT1is the productofthree positivedefinitematrices, thenT canbe expressed astheproduct ofthreepositivesemi-definitematricesaswell.Itwassuspectedthattheconverseofthis statementisalsotrue.However,thefollowingexampleshowsthatitisnotthecase.
Example1.1.LetT =
−9 −9
0 0
.ThenT = 9 3
3 2
13 −15
−15 18
1 1 1 1
istheprod- uct of three positive semi-definitematrices. However,T1 = [−9] is not the product of three positivedefinite matricesbecausedet(T1)<0.
Ofcourse,onemayimposetheobviousnecessaryconditionthatdet(T1)>0,andask whethertheconjectureisvalid withthisadditionalassumption.Nevertheless,itiseasy to modifyExample 1.1 byconsidering
T⊗I2=
−9I2 −9I2
02 02
= (A1⊗I2)(A2⊗I2)(A3⊗I3)
using thefactorization T =A1A2A3. Inthemodified example,wehaveT1=−9I2 and T2 = 02. By [1, Theorem 4], T1 = −9I2 is the product of no fewer than five positive definite matrices.
In thenextsection, we will giveacomplete characterizationfor those matricesthat canbewrittenastheproductofthreepositivesemi-definitematricesandnotfewer.Also, we add an easyto check necessary andsufficient condition forinvertible matrices that canbewrittenastheproductofthreepositivedefinitematrices.Withtheseresults,one canusetheJordanform ofagivenmatrixanditsnumericalrangetodecidewhetherit canbeexpressedastheproductoftwo,three,fourorfivepositivesemi-definitematrices.
2. Productof threepositivesemi-definitematrices
Wewill provethefollowing.
Theorem 2.1.Suppose T =
T1 R 0 T2
such thatT1 is invertible andT2 is nilpotent. (1)
Then T is a product of three positive semi-definite matrices if and only if one of the followingholds.
(a) R= 0 orT2= 0.
(b) R= 0,T2= 0,and T1 is theproduct ofthree positivedefinitematrices.
We establish some lemmas to prove Theorem 2.1. The first one is covered by [1, Theorem 1].
Lemma 2.2. Let A,S ∈ Mn such that S is invertible. Then A is a product of an odd numberof positivesemi-definite matricesif andonlyif S∗AS is.
Inthenextlemma,we needtheconceptof thenumericalrangeofamatrixA∈Mn
definedby
W(A) ={x∗Ax:x∈Cn, x∗x= 1}.
Thenumerical rangeis auseful tool instudying matrices. One maysee [2,Chapter 1]
forthebasicpropertiesofthenumericalrange.
Lemma2.3. SupposeT =
T1 R 0 0p
such that T1 ∈Mm isinvertible andR isnonzero.
ThenT isaproductof three positivesemi-definitematrices.
Proof. First,weshowthatthereisap×mmatrixS suchthatT1+RSis theproduct ofthreepositivedefinitematrices.
Ifm= 1,thereisS∈Cp suchthatT1+RS >0.Supposem>1 andRhasasingular valuedecomposition R =s1x1y1∗+· · ·+skxky∗k, where s1,. . . ,sk are nonzero singular valuesofR, andx1,. . . ,xk ∈Cmand y1,. . . ,yk ∈Cp are theircorrespondingright and left unit singular vectors accordingly. Let {e1,. . . ,em} be the standard basis for Cm. TakeaunitaryU suchthatU x1=e1.SinceT1 isinvertible,soisTˆ1=U T1U∗.Suppose ˆt∗1 is the first row of Tˆ1. Let v = ˆt1+e2 with >0 and Tˆ1() be the m×m matrix obtainedfromTˆ1 byreplacingitsfirstrowwithv∗.Take asufficientlysmall >0 such thatv isnotamultipleofe1 andthematrixTˆ1() isstillinvertible.
SetS=s−11 reiθy1v∗U withr >0 andθ∈[0,2π).Then
U(T1+RS)U∗=U T1U∗+U RSU∗= ˆT1+reiθe1v∗.
Sincev isnotamultipleofe1, therankonematrixe1v∗ isnotnormalandsoW(e1v∗) is an elliptical disk with foci 0 and v1 with length of minor axis
v2− |v1|2 > 0, wherev1 isthefirstentryofv;forexample,see[2,Theorem 1.3.6].Bythefactthatthe mapX →W(X) iscontinuous, forasufficientlylarger >0,W
e1v∗+e−iθTˆ1/r still contains0 asaninteriorpoint foranyθ∈[0,2π).Thensodoes
W
Tˆ1+reiθe1v∗
=reiθW
e1v∗+e−iθ r
Tˆ1
.
In addition, the value r can be chosen so thatr > |det( ˆT1)/det( ˆT1())|. Now by the linearity ofdeterminantwith respecttothefirstrow,
det(T1+RS) = det(U(T1+RS)U∗) = det( ˆT1+reiθe1v∗) = det( ˆT1) +reiθdet( ˆT1()).
Since |det( ˆT1)|<|reiθdet( ˆT1())|,there is θ∈[0,2π) such thatdet(T1+RS)>0.By [1,Theorem 3](seealsoProposition 2.4),T1+RSisaproductofthreepositivedefinite matrices.
Finally,note that T˜=
Im S∗ 0 Ip
T1 R 0 0p
Im 0 S Ip
=
T1+RS R
0 0p
.
By[3,Theorem 3.3],T˜isaproductofthree positivesemi-definitematrices,and soisT byLemma 2.2. 2
ProofofTheorem 2.1. SupposeT istheproductofthreepositivesemi-definitematrices.
IfR= 0 orT2= 0,thenwearedone.Else,T =T1⊕0p istheproduct ofthreepositive semi-definitematrices.By[3,Proposition 3.5],T1istheproductofthreepositivedefinite matrices.
To provetheconverse,weconsiderthefollowing threecases.
Case 1. Suppose R = 0. We use induction on p, the size of T2. If p = 1, the result follows from Lemma 2.3 as T2 = [0]. Assumethe resultholds for T2 with size at most p−1.Since T2 is nilpotent,withoutloss ofgenerality,wemayassumethatT2 isupper triangular with T2 =
T21 T22
0 0
, where T21 ∈ Mp−1.Write R = [R1 R2] where R1 is m×(p−1). If R1 = 0, then by induction, the matrix
T1 R1
0 T21
is a product of threepositivesemi-definitematrices,say,P1P2P3.Further,by[3,Theorem 2.2](seealso Proposition 3.1), we may assumethat both P1 and P2 are invertible. Let X =
R2
T22
with size(m+p−1)×1.Thenforany>0, P1 0
0 0
P2 P1−1X (P1−1X)∗ 1
P3 0 0 −1
=
P1P2P3 X
0 0
=
⎡
⎣T1 R1 R2 0 T21 T22
0 0 0
⎤
⎦=T.
Clearly,Q1=P1⊕[0] andQ3=P3⊕[−1] arepositivesemi-definitematrices.Nowone canchooseasufficientlysmall>0 sothatQ2=
P2 P1−1X (P1−1X)∗ 1
isalsopositive semi-definite.Thus,T isaproductofthree positivesemi-definitematricesQ1Q2Q3.
Now suppose R1 = 0.Then the (m+ 1)th columnof T is azerocolumn. Byinter- changingthe(m+1)thandthelastindices,onecanseethatT ispermutationallysimilar to
⎡
⎣T1 R˜ 0 0 T˜21 0 0 T˜22 0
⎤
⎦ where ˜R ism×(p−1), ˜T21 is (p−1)×(p−1), and ˜T22is 1×(p−1).
NoticealsothatR˜isnonzeroandT˜21isnilpotent.Byinduction,
T1 R˜ 0 T˜21
isaproduct of three positive semi-definite matrices P1P2P3. By [3, Theorem 2.2], we can further assumethatbothP2andP3 areinvertible.LetY = [ 0 T˜22] withsize1×(m+p−1).
Thenforany>0, P1 0
0 −1
P2 (Y P3−1)∗ (Y P3−1) 1
P3 0
0 0
=
P1P2P3 0
Y 0
=
⎡
⎣T1 R˜ 0 0 T˜21 0 0 T˜22 0
⎤
⎦.
Again,onecanchooseasufficiently small>0 suchthatallthree matricesinthe left sideoftheaboveequationarepositivesemi-definite.Thus,T ispermutationallysimilar toaproductofthreepositivesemi-definitematrices,andhenceT canalsobewrittenas aproduct ofthreepositivesemi-definitematrices.
Case 2.Suppose R = 0 andT2 isnonzero. Without loss of generality, we mayassume thatT2isuppertriangularwithzerodiagonalentrieswhilethefirstrowofT2isnonzero.
LetZ be thep×mmatrixwith1atthe(1,1)-entryandzeroelsewise.ThenZ∗T2= 0 andT2Z = 0.LetS =
Im 0 Z Ip
.Then
S∗T S=
Im Z∗ 0 Ip
T1 0 0 T2
Im 0 Z Ip
=
T1+Z∗T2Z Z∗T2
T2Z T2
=
T1 Z∗T2
0 T2
.
SinceZ∗T2= 0,theresultfollowsfrom Case 1andLemma 2.2.
Case 3.SupposeR= 0 andT2= 0.IfT1istheproductofthreepositivedefinitematrices, thenT =T1⊕0p istheproductofthree positivesemi-definitematrices. 2
Note that Theorem 2.1 depends on checking an invertible matrix is the product of threepositivedefinitematrices.Suchconditionsaregivenin[1,Theorem 3].Werestated theresult interms ofthe numericalrange inthe following proposition, whichis based onthediscussionin[1,Theorem 3andFact3.2].
Proposition 2.4. Let T ∈ Mn be such that det(T) >0. Then T is the product of three positivedefinitematrices ifandonly ifone ofthefollowingholds.
(a) W(T)contains 0asan interiorpoint.
(b) W(T)containsapositivenumber,andtheargumentsoftheeigenvaluesofT canbe arranged as: −π < θ1≤ · · · ≤θn< π such thatn
j=1θj = 0.
Note that in [1, p. 88], the author required in condition (3.6b), corresponding to our condition (b), that all real eigenvalues of T are positive, which is ensured by our assumption thatθj ∈(−π,π) forallj.
3. Determiningthe numberoffactors
In this section, we describe an algorithm to determine the smallest numberof pos- itive semi-definite matrices whose product equals a given A ∈ Mn with nonnegative determinant.
We first present the following theorem providingsome easy tests for amatrix A to be the product of two positive semi-definite matrices. The equivalence of conditions (a), (b), (c) weregiven in[3,Theorem 2.2].Weincludeashort proof,whichisdifferent from thatof Wu.
Proposition 3.1. LetA be asquarematrix. The followingareequivalent.
(a) A istheproductof twopositivesemi-definitematrices.
(b) A =BC,where B, C are positive semi-definite matrices such that B or C can be assumed tobeinvertible.
(c) A issimilar toanonnegativediagonal matrix.
(d) A is unitarily similar to an upper block triangular matrix such that the diagonal blocksare scalarmatricescorresponding todistinct scalars.
(e) The minimalpolynomialof A onlyhas simple nonnegativezeros.
Proof. Theequivalenceof(c),(d), (e)areclear.
(c) ⇒(b):A=S−1DS=S−1(S−1)∗(S∗DS)=S−1D(S−1)∗(S∗S).
(b)⇒(a): Trivial.
(a)⇒(c):SupposeA=BC,whereBandCaren×npositivesemi-definite.LetU be unitarysuchthatU∗BU =B0⊕0k,whereB0∈Mn−k ispositivedefinite.LetU∗CU = C11 C12
C21 C22
besuchthatC22∈Mk.AssumeV isunitarysuchthatV∗C11V =C0⊕0
forapositivedefinitematrixC0∈Mn−k−.WemayreplaceU byU(V⊕I) andassume thatC11=C0⊕0.BecauseCispositivesemi-definite,U∗CU =
⎡
⎣C0 0 C1
0 0 0
C1∗ 0 C22
⎤
⎦.Then
U∗AU=
B0 0 0 0k
⎡⎣C0 0 C1
0 0 0
C1∗ 0 C22
⎤
⎦=
B0 0 0 0k
⎡⎣C0 0 C1
0 0 0 0 0 0k
⎤
⎦.
Now B−
1 2
0 0
0 Ik
⎡⎣I 0 C0−1C1
0 I 0
0 0 Ik
⎤
⎦ B0 0
0 0k
⎡⎣C0 0 C1 0 0 0 0 0 0k
⎤
⎦
⎡
⎣I 0 −C0−1C1
0 I 0
0 0 Ik
⎤
⎦ B
1 2
0 0
0 Ik
=
B0−12 0 0 Ik
B0 0 0 0k
⎡⎣C0 0 0 0 0 0 0 0 0k
⎤
⎦
B012 0 0 Ik
=
B
1 2
0
C0 0 0 0
B
1 2
0
⊕0k.
Therefore,Aissimilarto
B012
C0 0 0 0
B012
⊕0k,whichispositivesemi-definiteand is(unitarily)similartoanonnegativediagonalmatrix. 2
Now,we areready to presentanalgorithm to checkwhetheramatrixA∈Mn with det(A) ≥ 0 can be written as a product of k positivesemi-definite matrices, but not fewer,fork= 1,2,3,4,5.
Algorithm3.2.LetA∈Mn withdet(A)≥0.
IfA=αInsuchthatα /∈[0,∞),thenAcanbeexpressedasaproductoffivepositive semi-definitematrices,butnotfewer. Otherwise,applyaunitarysimilarityto Ato get anuppertriangularmatrix
T =
T1 R 0 T2
such thatT1is invertible andT2is nilpotent.
(1) IfT isanonnegativediagonalmatrix,thenT isitselfapositivesemi-definitematrix.
(2) Condition (1)doesnothold,andAsatisfiesany oneoftheequivalentconditionsin Proposition 3.1.ThenAcanbeexpressed asaproductoftwopositivesemi-definite matrices,butnotfewer.
(3) Suppose (1) and (2) do not hold. Then A can be expressed as a product of three positivessemi-definitematrices, butnotfewer,ifanyof thefollowing holds.
(3.a) Ror T2 isnonzero.
(3.b) Both R = 0 and T2 = 0, and T1 is the product of three positive definite matrices.
In(3.b),the invertible matrixT1 is aproduct of three positivedefinite matrices if oneofthefollowingholds.
(i) det(T1)>0 andW(T1) contains0as aninteriorpoint,
(ii) 0 is not in the interior of W(T1) and W(T1) contains a positivenumber and θj = 0, where−π < θ1 ≤ · · · ≤θk < π arethearguments oftheeigenvalues ofT1.
(4) Suppose conditions (1), (2), (3) do not hold, i.e., T = T1⊕ 0p such that neither (i)nor(ii)holdsforthe uppertriangularmatrixT1. ThenA canbe expressed asa productoffourpositivesemi-definitematrices, butnotfewer.
Acknowledgements
The authors would like to thank the referee for his/her careful reading of the manuscript.TheresearchofCuiwassupportedbyNationalNaturalScienceFoundation ofChina11271217. LiisanaffiliatedmemberoftheInstituteforQuantumComputing, UniversityofWaterloo.HeisalsoanhonoraryprofessoroftheShanghaiUniversity,and the Universityof HongKong;hisresearch wassupportedbytheUSANSF grantDMS 1331021,theSimonsFoundationGrant351047,andtheNNSFgrantofChina11571220.
The research of Sze was supported by a HK RGC grant PolyU 502512 and a PolyU centralresearchgrantG-UC25.
References
[1]C.S.Ballantine,Productsofpositivedefinitematrices.IV,LinearAlgebraAppl.3(1970)79–114.
[2]HornJohnson,TopicsinMatrixAnalysis,CambridgeUniversityPress,Cambridge,1991.
[3]P.Y.Wu,Productsofpositivesemidefinite matrices,LinearAlgebraAppl.111(1988)53–61.
