 American Journal of Computational Mathematics, 2011, 1, 63-71 doi:10.4236/ajcm.2011.12007 Published Online June 2011 (http://www.scirp.org/journal/ajcm) Copyright © 2011 SciRes. AJCM The Conditions for the Con vergence of Power Scaled Matrices and Applications Xuzhou Chen1, Robert E. Hartwig2 1Department of Computer Science, Fitchburg St a t e U n i v er s i ty , Fitchburg, USA 2Department of Mathematics, North Carolina State University, Raleigh, USA E-mail: xchen@fitchburgstate.edu, hartwig@math.ncsu.edu Received February 28, 2011; revised May 30, 2011; accepted June 7, 2011 Abstract For an invertible diagonal matrix D, the convergence of the power scaled matrix sequence NNDA is investigated. As a special case, necessary and sufficient conditions are given for the convergence of NNDT, where T is triangular. These conditions involve both the spectrum as well as the diagraph of the matrix T. The results are then used to privide a new proof for the convergence of subspace iteration. Keywords: Convergence, Iterative Method, Triangular Matrix, Gram-Schmidt 1. Introduction The aim of iterative methods both in theory as well as in numerical settings, is to produce a sequence of matrices 01, ,AA, that converges to hopefully, something useful. When this sequence diverges, the natural question is how to produce a new converging sequence from this data. One of these convergence producing methods is to diagonally scale the numbers NA and form the sequence {}NNDA . Examples of this are numerous, such as the Krylov sequence (x, Ax, 2Ax,…), which when divergent can be suitably scaled to yield a dominant eigenvector. The convergence of power scaled iterative methods and more general power scaled Cesaro sums were studied by Chen and Hartwig [4,6]. In this paper, we continue our investigation of this iteration and derive a formula for the powers of an upper triangular matrix, and use this to investigate the convergence of the sequence {}NNnDT. We also investigate the subspace iterations, which has been started by numerous authorss [1,3,10,11,15], and turn our attention to the case of repeated eigenvalues. The main contributions of this paper are: •We present the necessary and sufficient conditions for convergence of power scaled triangular matrices {}NNnDT. We prove that these conditions involve both the spectrum as well as the digraph induced by the matrix T. •We apply the the convergence of power scaled triangular matrices with the explicit expression for the G-S factors of NNDT  and present a new proof of the convergence of simultaneous iteration for the case where the eigenvalues of the matrix A satisfy 12 1||| || |>||| |rr n   and ||=| |=ijij. Because of the explicit expression for the GS factors, and the exact convergence results, our discussion is more precise than that given previously [12,17]. One of the needed steps in our investigation is the derivation a formula for the powers of a triangular matrix T, which in turn will allow us to analyze the convergence of NNTDT. Throughout this note all our matrices will be complex and, as always, we shall use  and () to denote the Euclidean norm and spectral radius of (). This paper is arranged as follows. As a preliminary result, a formula for the power of an upper triangular matrix is presented in Section 2. It is shown in Section 3 that the convergence of NNTDT is closely related to the digraph induced by T. Section 4 is the main section in which convergence of general power scaled sequence NNDA is investigated and this, combined with path conditions in Section 3, is then used to discuss the convergence of NNDT. As an application we analyze the convergence results for subspace iterations, in which the eigenvalues are repeated, but satisfy a peripheral constraint. X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 64 2. Preliminary Results We first need a couple of preliminary results. Lemma 2.1. If 1<)(A and 0< <1i, then =0NkkkA (1) converges. Proof. For =0()= kkkfzz, we have =0 =0|()|=|| ||kkkkkfzzz. As the geometric summation on the right-hand side has radius of convergence 1, ()fz converges for all z such that ||<1z, which in turn tells us that the radius of convergence of ()fz is no less than 1. Therefore, from Theorem 6.2.8. of , ()fA converges. Next consider the triangular matrix 112 121,0=00nnnnuuuU, (2) which is used in the following characterization of the powers of a trangular matrix. Lemma 2.2. Let =000TaUcT where a and c are column vectors and suppose that =000NTNNNNNaUcNT. (3) then 11=0222=0 =0= NNk kNkNNkTkNki ikiaUc. (4) in particular, 1) if =0, then 212=0=NNTkNkNkaUc , (5) 2) if =0, then 212=0=NNTkNkNkaUc , (6) 3) if 0 and , then 121=01(/)=1(/) ,NNNkNkNNTkUac   (7) 4) if 0 and =, then 212=0=(1)kNNN TNkUNacNk  and (8) 5) if ==0, then 2=TNNaU c. (9) Proof. It is easily verified by induction that =NT 1NNNTyO, where 11=0==kkkkjTjTjkaUaOU OU1kT (10) and 111=0==NNNkkkNTcc  Ny. (11) Now 2121=0=0 12121=0=0 112212=0=0 =011=0= = =NkNkNkjT kNkjkNkNkNkNkjT kNkjkNkNNNkNkk NkjTkkkkjNNk kkaUcOUaUcUcaUcUc      Ny. Hence 12212=0=0 =012212=0=0 =021212=0=0 =0= = =NNNkNkkNkjT kkNkkjNNNkNkk TkNkjkkkjNjNNNkk TjNkjkkjkaUcaUcaUc       , completing the proof of (4). The special cases (1) - (5) are easy consequences of (4). Let us now illustrate how the power of T are related to its digraph. X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 653. The Digraph of T Suppose 0TaOU cOT is an (2)(2)nn upper triangular matrix. Correspondingly we select 2n nodes 01 1, ,,, nnSS SS, and consider the assignment 01 101112 121, 10000nnTnnnnnnSS SSSaSuuOucSSO (12) with 12=[, ,,]Tnaaa a and 12=[, ,,]Tnccc c. We next introduce the digraph induced by T, i.e. =(, )GVE where 01 1={, ,,}nVSSS is the vertex set and ={(, )|0}ijijESSt is the edge set. As usual we say (, )ijSS E if and only if 0ijt. A path from jS to kS in G is a sequence of vertices 1=,jrSS 2,, =rrklSSS with 1(, )rriiSS E, for =1, ,1il, for some l. If there is a path from jS to kS, we say that jS has access to kS and kS can be reached from jS. We write if (,),ifthereisapathfrom to ,if and ij ijij ijijijjiSS SSESS SSSSSSSS  Let 01 1=<, >={,,}npptSSSS be the sandwich set of 0S and 1nS, i.e., 1{,,}pptSS is the set of all the nodes from 1{, ,}nSS such that 0piSS 1nS, i.e., ipS can be reached from 0S and has access to 1nS. Let us now introduce the notation 1111=[,,] ,ˆ=,=[,,].TpptpptpptTpptaaaUUccc Then we have the following result. Lemma 3.1. cUaUca TT ˆ=. Proof. If 0jiji cua , then 0(, )iSS, (, )ijSS, (,jS 1)nSE, thus , ijSS, which implies that =1 =1==nnTiijj iijjij ijaUcaucauc . This completes the proof. □ This following corollaries are the direct consequences of the above lemma. Corollary 3.2. If 01nSS and there is no intermediate node that lies in 1{, ,}nSS on any path from 0S to 1nS, then 1) 10 nSS , i.e. 0, 2) 0=ˆ=cUacUa iTiT Corollary 3.3. cUacUa iTiT ˆ=, for =1, 2,.i We now turn to the main theorem of this section. Theorem 3.4. Let 112 11,0=00nnnnttt T be nonsingular and 1= ()= (,...,)ndiag TdiagTD. Then, the following statement are equivalent 1) NNTTD converges. 2) if ijSS , then ||>|| ji, i.e. if there is a path from iS to jS, then ||>|| ji. Proof. We prove the theorem by induction on n. For 2=n, 12=0T and 11/==01NN2NNNTMDT where 211 2121112[1(/)]/()if =/if =NNNN . It is easily seen that the convergence of (2)M implies that of NN1. Hence if 21 =, then 0=. Conversely, if 0, then 1|||>||p, and we are done. Subcase (b): There is no intermediate node between 0S and 1nS. In this case 10 nSS , and by Corollary 3.2., 0 and 0=ˆ=cUacUa iTiT for arbitrary i. Since the sandwich set  is empty, we see from Lemma 2.2., that [1(/)]/()if =/if =NNNN. (14) Now because we are given that NN converges and 0, we must have 1||| ji and assume that the hypothesis holds for matrices of size 1n or less. Since the graph condition also hold for 0{, , }nSS and 11{, ,}nSS, it follows by the hypothesis that all the entries in )(2NnM converges, with the possible exception of NN/. Consequently, all we have to show is that NN also converges, given the path conditions. Consider 12=022=01(/) 11(/)= 1/(1) =NiNiNTiNNiNTiUacifUNacNiif    (15) If 01nSS, then 10 nSS and therefore = 0. Moreover,  is empty and the right hand side of (15) is zero, i.e. 0=NN and we are done. So suppose 01nSS and thus ||>||. In this case 1(/)N converges (possibly to 0 when 0=). Now if π= then the second term of (15) vanishes by Lemma 2.2. Lastly suppose π , i.e. there are intermediate nodes 1,,pptSS. From Lemma 2.2., we recall that cUacUa iTiT ˆ= where 11*ˆ=ppppttSOSU. (16) Since for each i, 01pniSSS , we know that ||>||>||ip and thus )ˆ(|>|U. Hence  ˆ(/)<1U which implies that 12=0ˆˆiNTTiUUacaIc    . To complete the proof we observe that 12=0ˆiNiNiU also converges because of Lemma 2.1. with /ˆ=UA and iNi)/(=. We at once have, as seen in . Corolla ry 3.5. Let T be an upper triangular matrix and 1= ()= (,...,)ndiag TdiagTD. If 12| |>||>>||n, then NNTTD converges to an upper triangular matrix of diagonal 1. We now turn to the main result in this paper. Our aim is to characterize the convergence of NNAD in terms of the GS factorization of NA. 4. Main Theorem Let us denote the set of increasing sequences of p elements taken from (1, 2,,m) by , 11={=( ,,)|1<<}pm ppQIiii im and assume this set , pmQ is ordered lexicographically. Suppose ::=(, 1,...,)stss t is a subsequence of (1, 2,..., m) and we define 11:={=(,,)|<<<<}pppQstUuusuut. Clearly, , =1:pm pQQm. Suppose B is nm matrix of rank r. The determinant of a pp submatrix of A (1minp (, )mn), obtained from A by striking out pm rows and pn columns, is called a minor of order p of A. If the rows and columns retained are given by subscripts (see Householder ) 1, =( ,,)ppmIiiQ and 1, =( ,,)ppnJjjQ respectively, then the corresponding pp submatrix and minor are respectively denoted by IJA and )( IJAdet . The minors for which JI = are called the principal minors of A of order p, and the minors with ==(1, 2,,)IJp are referred to as the leading principal minors of A. Let 1, =( ,,)ppmIiiQ and 1, =( ,,)qqmJj jQ. For convenience, we denote by mpk QiI 1,][ the sequences of 1p elements obtained by striking out the kth element ki; while )( jI denotes the sequences of 1p elements obtained by adding a new element j after ki, i.e., 1()=(,,, )kIjiij. Note that if jip>, then )( jI is not an element of 1, pmQ because it is no longer an increasing sequence. If mqp, we denote the concaternation 11(,,, ,piij X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 67,)qj of I and J by IJ. It has qp  elements. Again, IJ may not be an element of , pqmQ. Since the natural sequence (1, 2,,p) of p elements will be used frequently, we particularly denote this sequence by =(1, 2,,)pp ; while []=(1,pt ,1, 1,,)tt p is simply denoted by  tp \. Next recall  that the volume )(BVol of a real matrix B, is defined as the product of all the nonzero singular values of B. It is known  that 2()=|det()|IJVol BB, (17) where IJB are all rr submatrices of B. In particular, if B has full column rank, then *()= det()VolBB B. (18) Lastly, suppose 12=[, ,,]rAaa a is an rn matrix of full column rank and =AYG (19) is its GS factorization so that the columns of 1=[ ,Yy 2,,]ryy are orthogonal and G is rr upper triangular matrix of diagonal 1. For rk , we define 1=[ ,,]kkAaa and =()kkVVolA. (20) It follows directly that 2*, =|det()|=det()IkkkkIQkmVAAA. (21) Theorem 4.1. Let A be an rn matrix of rank r and let YGA = be its GS factorization. Then () 2111, =det()det()/Ik Iklll lIQlnyAAV  (22) and *1( )2det( )=jjkjkjAAgV . (23) Proof. The result of (22) follows from Theorem 2.1. in , while on account of Corollary 2.1. in , *=(GY 1*)YYA. Hence we arrive at 2211*22=1*\ 21=1 =1== =(1)det()/njjjkj klj lkljjjnjt jtlt lkjjltVVgya yaVVaa AAV. Because lkltnlaa1= is just the (, tk) element of matrix AA*, we see that *\21=1 =1=(1)()det( )/jnjt jtjkltlkjjtlgaaAAV , which is the Laplace expansion along column j of *1( )det() jjkAA . Thus *1( )2det( )=jjkjkjAAgV , completing the proof. Remark: A different proof of (23) was given in [9,§1.4]. For a diagonal matrix 1=( ,...,)ndiag ddD, we say that D is decreasing, if 1||||ndd. (24) Moreover, D is called locally primitive, if it is decreasing and |=| |=ijijdd dd. (25) It is obvious that we can partition a decreasing matrix D as (1)( )=(,,)tdiag DDD (26) where each ()()1=(,,)ssiipssdiag eesD with 1||> 2||>>||t. As a special case, if D is locally primitive, then D can be written as 11=(,,)ptptdiag IID. (27) Now let us define jsjspq 0== (=1,st, 0== 00 pq) and 111:={=(,,)|uiiuu iQq qq  1<< <}uiq. Next, suppose rnNijN aA][= )( is a sequence of rn matrices and let () ()==[][]NNNNNij nrij rrAYG yg (28) be their GS factorization. Suppose B is a rn matrix, we can partition B conformally as D in (26). It is easily verified that the (, )uv element of (, )ij block ijB of B is exactly the 11(, )ijquqv element of the whole matrix B. B is said to satisfy condition () if for each uqki1= there exists iiuuqqQ :1 such that 1det 0qiukB . We now have the following theorem. Theorem 4.2. Let NA be a sequence of rn matrices of full column rank with GS factor NNNGYA =. Also suppose D is a diagonal matrix and rD is rr leading submatrix of D. Then 1) NNAD converges to B~ which satisfies condition ()  NG converges and NNYD converges to Z which satisfies condition () 2) If in addition D is decreasing, i.e. D satisfies (26), then for uqki1= and vql j1= (1ji ) 2()=NNjklNikyOd. (29) Proof. 1) The sufficiency is obvious. So let us turn to the necessary part. For 1=( ,...,)ndiag ddD, there exists a permutation Q such that DQQD*=ˆ is decreasing. Meanwhile, by hypothesis and the fact that =NNDA Q X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 68 ** ˆˆ()()=NNNNQDQ QAQDA with *ˆ=NNAQA , it follows that NNAD ˆˆ converges. So without loss of generality, we assume that D is decreasing and partition D as (26) and simply consider NNAD. We shall now, without risk of confusion, abbreviate the set 1111:={=(,,)|<<<<}uiiuu iuiQq qqq  to uQ and for ),...,(=1siiI set siiI dd1=. It at once follows that 1||=||qkiu. (30) We now have from (23) () *21()1(), 2, 1( )1111=det( ) /det( )det( )= (from Cauchy-Binet)det(() )()det()det()=()det()NkklN NklkIINkNklIQknINkIQknIINkNklIq QIq QiuiuNkIq QIq QiuiugAAVAAAAAA       2111()12112111() 1)det( )det( )=det(( ))det( )det( )() ()=Iqqiu iuNkNk lQIqQuui uqiuNkQIqQuui uNqqiuiu Nk lNk iNNQikkuuQuuAAAAAo       2211det( )()NqiuNk iNikAo ’ On account of (30), this is equal to 2111() 11122111det( )det( )()()det( )()Nqqiuiu Nk lNk iNNQiqquu iuiuNqiuNk iNQiquu iuAAoAo       . (31) Since NNAD convergence, so does the submatrices uiqINNAD 1)( and their determinant and hence 111111det( )=det[()()]() =det()qiuqqNNI iu iuNIqNiuqiuqNiuNIADADA converges, say, to uiqINAdet 1)~(. We have that consequently (31) converges to 111()21det( )det( )det( )qqiu iukklQuuqiukQuuAAA , in which the denominator is nonzero as A~ satisfies condition (). Hence NG converges and this implies that 1=NNNNNGADYD also converges. 2) Lastly, what remains is to show that NrN DY  converges if D is decreasing, i.e. D satisfies (26). Now for uqk i1=, vql j1= (1 ji ), it follows that ()1()1,21()1\\112111 11det( )det( )1=()det()det()1 =()det()det( ) =Ik INl NlNIQlnklNNkk lIk INl NlIq kQ Iq kQjv jvNIkNkIq QIq Qjv jvqjNlQvvAAydd VAAdAA      \() \111\1211111 11\() \11111det( )(det(()) )det( )det( )() ()1 =kk qkvjvNl Iq kQjvqNjvlNlQIqQvv jvqk kqkjv jvNl NlNNQIqllvv jNkAdAAAd       12111111 11det(( )))QvqjvNlNQIqQlvv jvA   X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 69\() \2111121\222111111 112det( )det( )()( )||=||det(( ))()||=|qk kqkNjvjvjNlNlNNNQillkvvlNNqjvkjNlNQjqvv jvNjiAAoddAo       \() \211111\() \11222111111 11() det()()()=|det(( ))()qk kqkNjvjv jNlNlNNQiqk kqkvv jvjvjNNqjv jNlNQjqvv jvdet AAoOAo        2Ni This completes the proof of 2). As a consequence of the above theorem we have Corollary 4.3. Suppose D is decreasing and NA's have orthogonal columns. If NNAD converges to B~ which satisfies condition (), then for uqk i1= and vql j1= (1 ji ) 2()=NNjklNikaOd Proof. In this case the GS factorization of NA are rNNIAA =. So the result is the direct consequnce of Theorem 4.2. Lemma 4.4. Suppose D is decreasing and NA's are of full column rank. If NNNpqN ADBB =][=)( converges, say, to B~, then NrN DA  converges iff 1) () (), , 11() =0=jjjjuvquq vuvjjBB (=1,j ,t) 2) If ji <, then ()( )()(), 11NijiNNiuvquq vijjBe converges. Proof. It is not difficult to see that the 11(, ijquq )v element of NrN DA  is , 11()( )()(), 11 ()=NNr q uqvijNijiNNiuvquq vijjADBe. (32) As () ,11NquqvijB converges and 1, it follows that (32) converges to zero in this case. Hence NrN DA  converges iff i) and ii) hold. Suppose B is an nn matrix and correspondingly there are n nodes 12, ,,nSS S. We say that B is indecomposable if for every i and j either ijSS or jiSS . Next we have Theorem 4.5. Let NA be of full column rank and NNN GYA= be its GS factorization. Suppose NNDA converges, say, to B~ which satisfies (). Then the following statement are true 1) If NrN DY  converges to 1=(,...,)sdiag ZZZ in which each block iZ~ (=1,,is) is indecomposable, then spssID=)( , =1, ,st 2) If spssID=)( , =1, ,st, then NrN DY  converges. Proof. From Theorem 4.2., the convergence of =NB NNDA implies the convergence of NG and NNYD. Suppose ZYD NN~. Then it follows, on account of Lemma 4.4, thatNrN DY converges to 1=(,,diag ZZ )sZ if a) () (), , 11() =0=jjjuvq uqvuvjjZZ , and b) if ji <, then ,11()( )(), 11 ()=()NNr q uqvijNijiNNiuvNq uqvijjYDDY e converges to zero. Now Corollary 4 says that for ji < () 2, 11, 111() ==NNquq vij jNNq uqvNij iquiYDY Od  and so b) is automatically statisfied in this case. Therefore NrN DY  converges iff a) holds. Since each iZ~ is indecomposable, for arbitrary (, )uv there exists a path either from uiqS1 to viqS1 or vice verse. In either case this implies that )()( =iviu for any u and v. We complete the proof of 1). 2) This time D is locally primitive, so we have )()(=jvju (=1,,jt) and hence X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 70 , 11()( )(), 11, 11 ()()if =()if =NNr q uqvijNijiNNiuvNq uqvijjNNq uq vjjYDDYei jDY ji By hypothesis, the above converges for ij=. The convergence for ji > is obvious; while the convergence for ji < can be easily achieved by noticing that () 2, 11, 111() ==NNquq vij jNNq uqvNij iquiYDY Od . Remark. From Theorem 4.5. we know that in the case of multiple eigenvalues, if uqk i1=, then 1(1)11()11[0,,0, ,,, 0,,0]qqqiiiNTkqqNiiiyppd . Let us now turn to the applications of this theorem. Our first application is the following result gives the general convergence result of power scaled triangular matrix. Corollary 4.6. Let D be diagonal and T be upper triangular. Then NNTD converges if and only if 1) Either 1|| |ijijSS . Proof. Let NNTA =. This time the GS factorization for NNTA = becomes )( NNTNTTDD and from Theorem 4.2., NNTD converges if and only if both NNTN TDG = and NTNDD converge. The convergence of NTNDD is equivalent to 1); while the convergence of NNTN TDG =, on account of Theorem 3.4., is exactly the same as the path condition 2). A relevant application of Theorem 3.4. is to the question of subspace iteration. Armed with Theorem 3.4. we can get a sharper theoretical result than was previously given. 5. Application to the Subspace Iteration Next, suppose T is an block upper diagonal matrix of the form 112112220=00ptpttptIT TITIT, (33) where |>||>|>|| 21 t. Let =()=diag TdiagTD 11(,,)ptptII and denote rrTrTDD )(= . Then from Theorem 3.4. it follows that NNTTD converges. Assume B is rn matrix of full column rank. Therefore itipnr1== and without loss of generality we can write wprisi1== for some 1spw . Thus we can write 111=(,,, )rpspswsTdiagII I. We now have Corolla ry 5.1 . Let T be nn upper triangular matrix defined as in (33), and let B be rn matrix whose columns are linearly independent. If NNNGYBT= is its GS factorization, then the followings hold 1) BTD NNT converges, say, to a limit A~. 2) NrTN DY  converges to 0P, where 1=(,PdiagP ,, )sPP and each iP (=1, ,is) is a ii pp matrix and P~ is a wps1 matrix. Proof. The result follows by simply choosing BTA NN= in Theorem 4.2. Let us now turn to the question of subspace iteration for a restricted class of matrices. Suppose that *=VTVA (34) is nn matrix, where V is unitary and T is as in (33). Then using the same iP as above we have Corollary 5.2. Suppose that A is an nn matrix which satisfies (34). let 0Y be an rn matrix whose columns are linearly independent and }{NY be sequence of matrices defined by the following factorization 0=NNNAYYG. Then 11 1[,,, ]NNTsssrYDVPVP VP. (35) Proof. Since 0=NNNAYYG, it follows that *0()=NNNVTV YYG. Partition 1=[ ,,]tVV V conformally to that of T in (33) and set 0*=YVB , then *=( )NNNTB VYG. (36) It is easily seen that the columns of NYV * are orthogonal. Therefore (36) can be regarded as the GS factorization of BTN. From Corollary 5.1., we have that for 1=[ ,,]tVV V *0NNTrPVYD, which is equivalent to X. Z. CHEN ET AL. Copyright © 2011 SciRes. AJCM 7111 1==[,,, ]0NNTrss srPYDVVPVPVP VP. 6. References  F. L. Bauer, “Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung Algebraischer Eigen- wertprobleme,” Zeitschrift für Angewandte Mathe- matik und Physik, Vol. 8, No. 3, 1957, pp. 214-235. doi:10.1007/BF01600502  A. Ben-Israel, “A Volume Associated with nm Matrices,” Linear Algebra and Its Applications, Vol. 167, No. 1, pp. 87-111, 1992. doi:10.1016/0024-3795(92)90340-G  X. Chen and R. E. 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