New Kronecker product decompositions and its applications

Transcription

1 RESEARCH INVENTY: Internatonal Journal of Engneerng and Scence ISBN: , ISSN: , Vol. 1, Issue 11 (December 01), PP New Kronecer roduct decomostons and ts alcatons Fuxang Lu Scence College and Insttute of Intellgent Vson and Image Informaton, Chna Three Gorges Unversty, Ychang, Hube, 44300, PR Chna Abstract: Frstly, two new nds of Kronecer decomostons s develoed,.e. KPGD and KPID; Secondly, the suffcent and necessary condtons and algorthms of Kronecer roduct(kpd), KPGD, and KPID are dscussed; At last, some useful roertes of the ran of the sum of Kronecer roduct gemel decomostons are obtaned. Keywords: Kronecer roduct decomoston, gemel decomoston, somer decomoston, ran, dmensonalty reducton. SC: 65L09,65Y04,15A69 I. Introducton Because of ts elegant algebrac roertes, the Kronecer roduct s a useful tool to solve matrx equatons and the nearest ronecer roduct roblems[1,], do nference n multvarate analyss[3], and construct fast and ractcal algorthms n sgnal rocessng, mage rocessng, comuter vson, semdefnte rogrammng, quantum comutng, lnear systems and stochastc automata networs etc. see[1,4,5,6,7,8,9,10] and so on. eantme, the alcatons n nearly all those areas are related to some certan nds of Kronecer roduct decomostons whch n fact are the nverse roblems of Kronecer roduct, see [3,13,14,18] etc. Usually, Kronecer roduct decomoston(kpd) means that a matrx can be transformed to the ronecer roduct form of the other matrces A;B, :e: = A B; Kronecer roduct gemel decomoston(kpgd) means the Kronecer roduct wth the secal case A = B; And Kronecer roduct somer decomoston(kpid) corresonds to the case = A A. Obvously, these decomostons often have many solutons as well as the other nverse roblems. In ths drecton, Eugene Tyrtyshnov[14] has some nterestng wor about Kronecer rans; T.G.Kolda[1] has some meanngful wor on orthogonal tensor decomostons; DE Launey and Seberry[18] develoed some roertes and ther alcatons on the strong Kronecer roduct; In addton, Sadegh Joar and oler ehrmann[11], Jun-e Feng, James Lam, Ymn We[13] etc, have obtaned some useful roertes of the sum of Kronecer roducts. Undoubtedly, these dfferent decomostons are helful for dmensonalty reducton rocedure whch s very mortant ey for hgh dmensonal mage rocessng and gene analyzng. Unfortunately, many natural questons about seemngly smle cases are stll not answered n ste of an ever ncreasng nterest and some sgnfcant results wth alcatons, such as the condtons and the ran of the sum of these decomostons. In ths aer, from the ersectve of the nverse roblem theory, we manly exlore the suffcent and necessary condtons and algorthms of KPD, KPGD, KPID, and obtan some useful roertes of the ran of the sum of Kronecer roduct gemel decomoston. And our research wors are manly motvated by dong multvarate statstcal nference and huge dmensonal statstcal analyss, solvng hgh dmensonal matrx equatons and constructng the algorthm of mage rocessng and comuter vson. 5

2 New Kronecer roduct decomostons and ts alcatons II. KPD, KPGD And KPID Problems Obvously, the essental recondtons of KPD( = A B), KPGD( = A A) and KPID( = A A ) means that the matrces,a,b have the roer columns and rows. And ths s easly verfed, so, we assume that all the matrces n the followng dscusson have the sutable column and row numbers..1 The suffcent and necessary condtons of KPD n m A ( a,, a ) R n wth a R,1 m, then denotevec( A) ( a1', a',, a m ')'. Let 1 m Frstly, we exlore the suffcent and necessary condtons of Kronecer roduct decomoston, and gve the elegant form of ths result as follows. Theorem.1. (KPD) For an arbtrary matrx mr ns R, 11 1n m1 mn r s, ( R, 1,, m, j 1,, n), can be decomosed to the form A B, where m n A R r s, B R,m, n, r, s are some certan ntegers. (equvalent to) ran vec( ), vec( ),, vec( ),, vec( ) = n raculously, the roof of ths theorem s not dffcult, so the detals of the roof are omtted. Remar 1. Generally seang, the KPD of an arbtrary matrx s not unque, because of (A) (lb) = A B wth l = 1 for arbtrary constants ; l and matrces A,B. The followng algorthm descrbes the general rogram of KPD roblem that ncludes whether a matrx can be decomosed or not and how to get the results of KPD. Algorthm 1(KPD): ste 1: nut, m, n, r, s, verfy the sze of s mr ns and! = 0 mn ste : defne, 1,, m, j 1,, n ste 3: calculate vec( ) ste 4: f ran{ vec( 11), vec( 1),, vec( 1n ),, vec( mn) } == 1 goto ste 5 else outut can not decomoston ; end ste 5: loo for the frst! = 0, defne B = ste 6: calculate a : vec( ) = a vec( B ), 1,, m, j 1,, n ste 7: defne A ( a ), 1,, m, j 1,, n; outut A,B; end 6

3 New Kronecer roduct decomostons and ts alcatons. The suffcent and necessary condtons of KPGD and KPID Wth the smlar nference, we can have the followng condtons about KPGD. Theorem.. (KPGD) For an arbtrary matrx 11 1q 1 A A, where q q R ( 0 ),, ( R q, 1,,, j 1,, q), can be decomosed to the form q A R, q, are some certan ntegers. (equvalent to) there exsts a subbloc 0, and m, 0 where ( mst, ) 1,,, 1,, f denote / m, ( al ) 1,,, l1,, q Corollary.3. Denote j s t q, A or A, then for,, l l al A. q and ( mst, ) s1,,, t1,, q ran q R R ( 0 ), ( ), 1,,, j 1,, q,, vec( ), vec( ),, vec( ),, vec( ) = q, then the matrx can be carred out KPGD ( AA ) The followng algorthm descrbes the general rogram of KPGD roblem that ncludes whether a matrx can be decomosed or not and how to get the results of KPGD. Algorthm (KPGD): q ste 1: nut,, q, verfy the sze of s q and! = 0 ste : defne flag=0,, 1,,, j 1,, q ste 3: for (, j ), 1,,, j 1,, q f: == 0, contnue; else f:! = 0&& m, 0 flag=1; brea; j else: defne /, flag=; brea; B m ste 4: f flag==&& == B B A = B; outut A; end else outut can not gemel decomoston ; end Theorem.4. (KPID) For an arbtrary matrx q R ( 0 ), 11 1q 1 q q, ( R, 1,,, j 1,, q), can be decomosed to the form 7

4 New Kronecer roduct decomostons and ts alcatons A A', where q A R, q, are some certan ntegers. (equvalent to) there exsts a subbloc 0, and m, 0 where ( mst, ) 1,,, 1,, j s t q, such that, l, l al A', where A ( al ) 1,,, l1,, q, and A' / m, or, A' / m. 1,,, j 1,, n, then III. The Ran Of The Sum Of KPGD In ths secton, we dscuss some roertes of the ran of the sum of KPGD( A 1 A ). Lemma 3.1. Let A and B be n n real symmetrc matrces. () There exsts a real orthogonal matrx Q such that Q AQ and Q BQ are both dagonal f and only f AB = BA (that s AB s symmetrc). () The revous result holds for more than two matrces. A set of real symmetrc matrces are smultaneously dagonalzable by the same orthogonal matrx Q f and only f they commute arwse. see George A. F. Seber[17] for more detals about matrces smultaneous dagonalzaton. Theorem 3.. Let A1,, A ( ) be n n real symmetrc and ostve defnte(> 0) or negatve defnte(< 0) matrces. If they commute arwse, then ran( A 1 A ) = ran( A ) n 1. Proof: By the result of Lemma 3.1, there exsts an orthogonal matrx Q, such that QA Q dag, 1,,, where 1,, ' ( 1,, n) n are the egenvalues of ( Q Q )( )( ) ( 1 1 1,, n) ( 1,, n) A A Q Q dag dag Obvously, the A, and 0, ran ( ( 1 1,, n) ( 1,, n)) dag dag n, then the roof s comleted. Wth smlar dscussons, we have the followng two theorems. Theorem 3.3. Let A1,, A ( ) be n n real symmetrc matrces, there at least a ostve defnte(> 0) or negatve defnte(< 0) matrces. If they commute arwse, then ran( A 1 A ) = ran( A ) n 1. Theorem 3.4. Let A1,, A ( ) be n n real symmetrc and ostve defnte(> 0) or negatve defnte(< 0) matrces. If they commute arwse, then ran( ) max ran ( A1 A1 ),, ran ( A A ) A 1 A. 8

5 New Kronecer roduct decomostons and ts alcatons These results wll be helful to study the solutons of the followng general Sylvester matrx equaton roblem[19, 0]: A XA A XA C A A vec( X ) vec( C) Theorem 3.5. Let, AB be n n real symmetrc matrces wth AB BA 1,, n, then the egenvalues of B are 1,, n n, the egenvalues of A are, defne the vector ( 1,, n )', ( 1,, )', ( 1,, n )', where 1,, n s an arbtrary ermutaton of the elements ( 1,, )', Q j j j j n n max ( ) (, ) : 0,, 1,,, the number of Q, then n ran AAB B n mn Proof: There exsts an orthogonal matrx Q, such that QAQ ' dag( 1,, n ) and QBQ dag n ' ( 1,, ). Thus, ( Q Q)( A A B B)( Q ' Q ') dag( 1,, n) dag( 1,, n) dag( 1,, n) dag( 1,, n). Calculate the number of nonzero elements of j j,, j 1,, n, and the result s obtaned. IV. Dscusson In ths aer, we dscuss the suffcent and necessary condtons and algorthms of KPD, KPGD and KPID roblems, whch lay a great role n all nds of Kronecer roduct alcaton areas, and obtan some useful roertes of the ran of the sum of KPGD( A 1 A ) n smultaneous dagonalzaton stuaton whch erforms some wonderful algebra advantages. ore nterestng wor n the future maybe nclude the followng asects: the condtons that a matrx can be decomosed to the form A 1 B whch s a meanngful wor esecally n sarse matrces cases[11, 1], decomosng rogram, and the roertes of the ran of the sum of decomoston n more general cases. Also, those Kronecer roduct decomoston roertes maybe assocated wth seeng the suffcent and necessary condtons under whch the followng matrx equaton n A ( ) 1 I X B C has a unquely soluton, where A, B and matrces, and X s an unnown matrx. C are nown V. Acnowledge Ths wor s suorted by Natonal Natural Scence Foundaton of Chna (NSFC) Grants

6 New Kronecer roduct decomostons and ts alcatons References [1] C.F. Van Loan, The ubqutous Kronecer roduct, Journal of Comutatonal and Aled athematcs, 13(000) [] A.Wu, G.Duan, Y.Xue, Kronecer mas and Sylvester-olynomal matrx equatons, IEEE Trans, Automat. Control 5(5)(007) [3] Hu,J, Fuxang Lu, and S. Ejaz Ahmed, Estmaton to arameters n the growth curve model wthout assumton of normalty va analogy and least squares, submtted to J. ultvarate Anal. [4] A. Barrlund, Effcent soluton of constraned least squares roblems wth Kronecer roduct structure, SIA J. atrx Anal. Al.19(1998) [5] B.J. Le E.A. Hendrs, ult-ste Smultaneous ultle Camera Calbraton, ASCI,. 06, 003. [6] D.W. Fausett, C.T. Fulton, Large least squares roblems nvolvng Kronecer roducts, SIA J. atrx Anal. 15(1994) 19-7 [7] J. Granata,. Conner, R. Tolmer, Recursve fast algorthms and the role of the tensor roduct, IEEE Trans. Sgnal Process. 40(199) [8] P.A. Rgeala, S. tra, Kronecer roducts, untary matrces, and sgnal rocessng alcatons, SIA Rev. 31(1989) [9] N.P. Ptsans, The Kronecer roduct n aroxmaton, fast transform generaton, Ph.D. Thess, Deartment of Comuter Scence, Cornell Unversty,1997. [10] Amy N.Langvlle, Wllam J.Stewart, The Kronecer roduct and stochastc automata networs, Journal of Comutatonal and Aled athematcs,167 (004) 49C447 [11] Sadegh Joar, Voler ehrmann, Sarse solutons to underdetermned Kronecer roduct, Lnear Algebra and ts Alcatons, 431(009) [1] T.G.Kolda, Orthogonal tensor decomostons, SIA J.atrx Anal. Al.3(1)(001) [13] Jun-e Feng, James Lam, Ymn We, Sectral roertes of sums of certan Kronecer roducts, Lnear Algebra and ts Alcatons, 431(009) [14] Eugene Tyrtyshnov, Tensor rans for the nverson of tensor-roduct bnomals, Journal of Comutatonal and Aled athematcs, 34(010) [15] J..Ford, E.E.Tyrtyshnov, Combnng Kronecer roduct aroxmaton wth dscrete wavelet transforms to solve dense, functon-related systems, SIA J.Sc.Comut,5(3)(003), [16] aro Huhtanen, Real lnear Kronecer roduct oeratons, Lnear Algebra and ts Alcatons, 418(006) [17] George A. F. Seber, A atrx Handboo for Statstcans, P345, 007, Wley,New Yor. [18] DE Launey and Seberry, The strong Kronecer roduct, 1994, Journal of Combnatoral Theorey, Seres A, Vol. 66, No., ay 1994, [19] P. Lancaster, Exlct soluton of lnear matrx equatons, SIA Rev. 1 (1970) [0] W.J. Vetter, Vector structures, solutons of lnear matrx equatons, Lnear Algebra Al. 10 (1975) [1] Xu Juny, Sun We, Q Dongxu, Kronecer Products and Alcatons, Journal of Comuter-Aded Desgn & Comuter Grahcs. Vol.15, No.4, Ar. 003,