The finite-step realizability of the joint spectral radius of a pair of $dtimes d$ matrices one of which being rank-one

The finite-step realizability of the joint spectral radius of a pair of d × d matrices one of which being rank-one ✩ Xiongpin g Dai a , Y u Huang b , Jun Liu c , Mingqing Xiao c a Department of Mathemat ics, Nanjing Universi ty , Nanjing 210093, P eople’ s Republic of China b Department of Mathemat ics, Zhongshan (Sun Y at-Sen) Unive rsity , Guangzhou 510275, P eople’ s R epublic of China c Department of Mathemat ics, Southern Illinois Universit y , Carbondale, IL 62901-4408, USA Abstract W e stu dy the finite-step realizab ility of the joint / gen eralized spectral radius o f a pair of re al d × d matrice s { S 1 , S 2 } , one of which h as rank 1, where 2 ≤ d < + ∞ . Let ρ ( A ) deno te the spectral radiu s of a square matrix A . Then we p rove that there always exists a finite-length word ( i ∗ 1 , . . . , i ∗ ℓ ) ∈ { 1 , 2 } ℓ , for some finite ℓ ≥ 1, such that ℓ q ρ (S i ∗ 1 · · · S i ∗ ℓ ) = sup n ≥ 1  max ( i 1 ,..., i n ) ∈{ 1 , 2 } n n p ρ (S i 1 · · · S i n )  ; that is to say , there holds the s pectral finiteness property for { S 1 , S 2 } . This implies that stability is algorithm ically decidable for { S 1 , S 2 } . K eywor ds: Joint / generalize d spectral radius, rank-on e matrix, finiteness conjecture, exponential stability . 2010 MSC: Primary 15B52; Secondary 15A60, 93D20 , 65F15. 1. In troduction Let S = { S 1 , . . . , S K } ⊂ R d × d be an arbitrary fin ite s et of real d -by- d matrices and k · k a matrix norm o n spa ce R d × d , whe re 2 ≤ d < + ∞ and K ≥ 2. T o capture th e max imal growth rate of th e trajectories gene rated b y rando m prod ucts o f matrices S 1 , . . . , S K in S , in 1 960 [ 55 ] G.-C. Rota and G. Strang introdu ced the v ery important concept – joint spectr al radius of S – by ˆ ρ ( S ) = lim n → + ∞  max ( i 1 ,..., i n ) ∈ K n n p k S i 1 · · · S i n k  . Here K n : = n -time z }| { { 1 , . . . , K } × · · · × { 1 , . . . , K } stands for the set of all words ( i 1 , . . . , i n ) of fi nite-leng th n , composed by the letters 1 , . . . , K , for any integer n ≥ 1. Let Σ + K = { i · : N → K } , where N = { 1 , 2 , . . . } , ✩ This work was supported in part by Nationa l Science Foundatio n of China (Grant Nos. 11071112 and 11071263) and in part by NSF 1021203 of the United States. Email addr esses: xpdai@nju.edu.c n (Xiongping Dai), stshyu@mail. sysu.edu. cn (Y u Huang ), jliu@mat h.siu.edu (Jun Liu), mxiao@math.s iu.edu (Mingqing Xiao) Prep rint submitted to Linear Algeb ra and its Applications Nove mber 21, 2018 be the set o f all one-sided in finite sequen ces ( also called switching signals of S ) . Then we see, from N. Barabanov [ 1 ] for example, that ˆ ρ ( S ) < 1 if and on ly if k S i 1 · · · S i n k → 0 as n → + ∞ ∀ i · ∈ Σ + K . In other words, ˆ ρ ( S ) < 1 if and only if the linear switched dynam ical system, also write as S , x n = x 0 · S i 1 · · · S i n , x 0 ∈ R d and n ≥ 1 , i · ∈ Σ + K , it is absolutely asymptotically stable, where the initial state x 0 ∈ R d is though t o f as a r ow vector . In fact, from [ 16 ] there follows ˆ ρ ( S ) = max i · ∈ Σ + K  lim sup n → + ∞ n p k S i 1 · · · S i n k  . So, ˆ ρ ( S ) is a no nnegative real nu mber which is indepen dent of the norm k · k used here. It is a well-known fact that the jo int spectral radius ˆ ρ play s a critical ro le in a variety o f a pplications such as switched dy namical systems [ 1 , 35 , 4 , 28 , 60 , 61 , 3 , 23 , 43 , 44 , 5 7 , 31 , 15 , 1 3 ], d i ff erential equations [ 2 , 24 , 13 ], coding theory [ 50 ], wa velets [ 17 , 18 , 30 , 47 ], c ombinato rics [ 19 ], an d so on. Although ˆ ρ ( S ) is indepen dent of the norm k · k used here, its ap proxim ation b ased o n the ab ove limit d efinition d oes rely upon an explicit cho ice of the n orm k · k and has been a substantially interesting topic , for examp le, in [ 35 , 42 , 2 2 , 52 , 53 , 59 , 4 7 , 6 , 3 4 , 51 , 38 , 39 , 40 , 41 , 4 9 ]. In general, computin g ˆ ρ by d efinition ca nnot stop at som e finite- time n , as sh own by the single matrix A =  1 0 1 1  where ˆ ρ ( A ) = lim n → + ∞ n √ k A n k = 1 b y the classical Gel’fand spectral radiu s fo rmula, h owe ver, there hold s n √ k A n k > 1 fo r all n ≥ 1. For that reason in part, I . Daubechies and J. Lagarias in 1992 [ 17 ] defined the equally impor tant c oncept – generalized spectr al radius o f S – by ρ ( S ) = lim sup n → + ∞  max ( i 1 ,..., i n ) ∈ K n n p ρ (S i 1 · · · S i n )  , where ρ ( A ) stands for the usua l spectral r adius fo r any matrix A ∈ R d × d . An d they con jectured there th at a Gel’fand- type formula sh ould ho ld for S. This was proved by M.A. Berger an d Y . W ang in 1 992 [ 4 ], i.e., there holds the following Gel’fand -type formula. Berger -W ang Formula 1.1. ρ ( S ) = ˆ ρ ( S ) , for any bo unded s ubset S ⊂ R d × d . Because of its importanc e, this Gel’fand -type spectr al-radius formu la has been reproved by using di ff er ent interesting ap proache s, for example, in [ 20 , 56 , 1 0 , 8 , 1 2 ]. Accord ing to this for- mula, the comp utation of ρ ( S ) becom es an impo rtant subject at once, which leads to the fo llowing significant prob lem m otiv ated by ρ ( A ) = n p ρ ( A n ) for any square matrix A . Problem 1.2 (Spectr al Finiten ess Property ) . Does there exist any wor d ( i 1 , . . . , i n ) of fi nite-length n ≥ 1 such that ρ ( S ) = n p ρ (S i 1 · · · S i n ) , for any S = { S 1 , . . . , S K } ⊂ R d × d ? 2 This spec tral finiteness pr operty m eans th at ρ ( S ) is co mputation ally e ffi cient. It was c onjec- tured, respectively , by E.S. Pyatn itski ˇ i [ 54 ] f or its continu ous-time version, I. Daub echies and J. Lagar ias in [ 17 ], L. Gurvits in [ 28 ], a nd by J. Lagarias and Y . W ang in [ 4 2 ]. If this were true for S, then f rom th e Berger-W ang f ormula, it f ollows that we would realize e ffi ciently the joint / gen eralized spectral radius ˆ ρ ( S ) only by computation of finite steps, and the in terests w o uld arise from its connectio n with the stability question for S . Unfortu nately , this imp ortant “spectra l finiteness co njecture” has been disproved by T . Bousch and J. Mairesse in [ 9 ] usin g measure-th eoretical id eas, also r espectively by V . Blonde l et al. in [ 7 ] exploiting combin atorial proper ties of perm utations of produ cts of positive matrices, an d by V . K ozyakin [ 36 , 3 7 ] emp loying the th eory of dynam ical systems, all o ff e red the existence of counterexam ples in the case wh ere d = 2 and K = 2. Moreover , an explicit expression f or s uch a counterexam ple has been fou nd in the recent work of K. Hare et al. [ 29 ]. Although the finiteness conjecture fails to exist, the id ea of Problem 1.2 is still to be very attractive and impor tant due to developing e ffi cient algorithms because the computatio n of the joint s pectral radius ˆ ρ m ust be implemen ted in finite a rithmetic. So me c onjectures in sp ecial case still keep open , f or example, in M. Maesumi [ 46 ] and R. Jun gers and V . Blondel [ 3 2 ]. Many positive e ff orts have been made and studies show that spectral finiteness pro perty m ay be true in a nu mber o f interesting cases, f or exam ple, see [ 28 , 42 , 23 , 58 , 5 , 32 , 25 , 26 , 27 , 33 , 11 , 48 ], including th e case were the matrices S 1 , . . . , S K are symmetric, or if the Lie alg ebra associated with the set of matrices is solvable [ 58 , Cor ollary 6 .19]; in this case ρ ( S ) = max 1 ≤ i ≤ K ρ (S i ), see [ 2 8 , 4 5 , 32 ]. Particularly , in A. Cicon e et al. [ 11 ] it was proved, b ased on R. Jung ers and V . Blondel [ 32 ] which is for all pair s of 2 × 2 binary matr ices, that every pairs of 2 × 2 sign- matrices S 1 , S 2 have th e spectral finiteness proper ty d escribed in Problem 1.2 . In the presen t p aper, based on th e impo rtant work of Barabanov [ 1 ], we will prove, mathe - matically in Section 2 and numeric ally in Section 3 , the following finiten ess result. Theorem 1. 3. Let 2 ≤ d < + ∞ and S = { S 1 , S 2 } be an arbitrary pair of real d × d matrices. If one of S 1 and S 2 has rank 1 , then S has the spectral finiteness pr operty . This m eans, from [ 32 , Pr oposition 1], that stability is alg orithmically d ecidable, for every pairs of re al d × d matr ices S 1 , S 2 if one o f wh ich has rank 1. If , in addition, S is irreducible, then S possesses the rank one property intr oduced by I .D. M orris in [ 48 ]. Ho wev er , the counterexamp le of Har e et al. [ 29 ] shows that Morris’ s rank one proper ty is neither necessary nor su ffi cien t for the finiteness prop erty . So, our rank 1 cond ition describe d in Theo rem 1.3 is substantial for our statement. By S + , it means for the multiplicative semigr oup generated by S 1 , S 2 , i.e, S + = G n ≥ 1  S i 1 · · · S i n | ( i 1 , . . . , i n ) ∈ K n  where K = { 1 , 2 } . Here F means the disjoin t un ion. T o mathem atically prove Th eorem 1.3 , our appr oach is to consider its equivalent statement form ulated as follo ws: Theorem 1.4. Let S = { S 1 , S 2 } be an arbitrary pa ir of real d × d matrices, one of which ha s rank 1 . I f ρ ( A ) < 1 for all A ∈ S + , then ρ ( S ) < 1 ; namely , th e in duced switched dyn amics S is absolutely asymptotically and exponentially stable. This stability was first con jectured by E.S. Pyatnitski ˇ i in 19 80s, see, e.g., [ 54 , 28 , 57 ] and it has been the subject of substantial recent research interest, for example, in [ 28 , 56 , 57 , 15 , 13 ]. 3 This pap er is organ ized as follows. In Section 2 , we will provid e th e proo f of o ur main results. Moreover , we a lso give an explicit formu la for the co mputation o f the gen eralized spectral radius. Sev eral examples a re p rovided in Section 3 to illustrate the theoretical outcom es. The paper e nds with conclu ding r emarks in Section 4 . 2. Finiteness property of a pair of d × d mat rices This section is dev oted to mathematica lly proving o ur main results, Theorems 1.3 and 1.4 . T o p rove Theorem 1.3 , we first prove the following absolute stability theorem, which is im- portant not just to o ur sp ectral finiteness theo rem, but also to the stabilizability o f switched dy- namical systems [ 15 , 57 ]. Theorem 2.1. Let S = { S 1 , S 2 } ⊂ R d × d , 2 ≤ d < + ∞ , be p eriodically switched stable; that is to say , ρ ( A ) < 1 ∀ A ∈ S + . Then, if one of S 1 , S 2 is of rank 1 , S is absolutely e xpo nentially stable, i.e., k S i 1 · · · S i n k exponentially fast − − − − − − − − − − − − → 0 a s n → + ∞ , i.e., lim sup n → + ∞ 1 n log k S i 1 · · · S i n k < 0 , for all switching signals i · : N → { 1 , 2 } . Here S + is the multiplicati ve semigrou p generated by S as described in Section 1 . Recall that S is said to be irreducible, provided that th ere is no com mon, no ntrivial and pro per inv arian t linear subspaces of R d , for S 1 , S 2 . The following result ho lds trivially b y induction on d together with th e Berger-W ang f ormula, which is a standard result in the theor y o f linear algebras. Lemma 2.2 ( See, e .g.,[ 1 , 4 , 12 ]) . F o r any S = { S 1 , S 2 } ⊂ R d × d , ther e e xists a nonsingula r matrix P ∈ R d × d and r p ositive inte gers d 1 , . . . , d r with d 1 + · · · + d r = d such that PS i P − 1 =      S (1 , 1) i 0 d 1 × d 2 · · · 0 d 1 × d r S (2 , 1) i S (2 , 2) i · · · 0 d 2 × d r . . . . . . . . . . . . S ( r , 1) i S ( r , 2) i · · · S ( r , r ) i      ( i = 1 , 2) , wher e S ( k ) : = n S ( k , k ) 1 , S ( k , k ) 2 o ⊂ R d k × d k is irr edu cible for each 1 ≤ k ≤ r , such that max  ρ ( S ( k ) ) : 1 ≤ k ≤ r  = ρ ( S ) = ˆ ρ ( S ) = max  ˆ ρ ( S ( k ) ) : 1 ≤ k ≤ r  . When S is itself irreducib le, r = 1 in Lemma 2.2 . The following impo rtant theo rem, due to N. B arabanov , is extreme ly valuable to th e proof o f Theorem 2.1 . Barabanov’s norm theorem 2 .3 ( See [ 1 ], also [ 60 , 12 ]) . I f S = { S 1 , S 2 } ⊂ R d × d is irr edu cible, then ther e is a vector norm | | | | · | | || ∗ on R d such that ther e hold the following two statements. (1) ˆ ρ ( S ) = max  n p | | | | S i 1 · · · S i n | | || ∗ : ( i 1 , . . . , i n ) ∈ K n  for all n ≥ 1 . 4 (2) T o any ˆ x ∈ R d , there co rr espon ds an infinite sequ ence, say i · ( ˆ x ) : N → K , satisfying th at | | | | ˆ x · S i 1 ( ˆ x ) · · · S i n ( ˆ x ) | | || ∗ = ˆ ρ ( S ) n | | | | ˆ x | | || ∗ for all n ≥ 1 . Her e K = { 1 , 2 } and the matrix norm | | | | · | | || ∗ on R d × d is naturally in duced by the norm | | | | · | | || ∗ on R d . Using the Berger-W ang formula, Lemma 2.2 and Barab anov’ s norm theorem, we now can prove Theorem 2.1 . Proof of Theorem 2.1 . Let S = { S 1 , S 2 } ⊂ R d × d with ran k(S 2 ) = 1. Th en accordin g to Lem ma 2.2 , there is no loss of generality is assuming th at S is irr educible. Since S is period ically switch ed stable, we h a ve ˆ ρ ( S ) ≤ 1 by the defin ition of ρ ( S ) and the Berger-W ang fo rmula. T herefor e, from Barabanov’ s th eorem, it f ollows that there exists a vector n orm || | | · | | || ∗ on R d , wh ich induces a matrix norm , write als o | | | | · | | || ∗ , on R d × d such that | | | | S 1 | | || ∗ ≤ 1 and | | | | S 2 | | || ∗ ≤ 1 . W e simply write S 1 =  a i j  d × d and S ℓ 1 = ℓ -time z }| { S 1 · · · S 1 = h a ( ℓ ) i j i d × d ∀ ℓ ≥ 1 . As S 2 is of rank 1, it f ollows, f rom th e Jo rdan c anonical for m theo rem, th at ther e is no loss of generality in assuming that S 2 = B 1 : =  λ 0 1 × ( d − 1) 0 ( d − 1) × 1 0 ( d − 1) × ( d − 1)  where 0 < | λ | < 1 or S 2 = B 2 : =        0 0 0 · · · 0 1 0 0 · · · 0 0 0 0 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 0        . As { S 1 , S 2 } is per iodically switched stable, it follows from the classical Gel’fand spectral radiu s formu la that | | | | S n i | | || ∗ exponentially f ast − − − − − − − − − − − − → 0 as n → + ∞ , i.e., log ρ (S i ) = lim n → + ∞ 1 n log || | | S n i | | || ∗ < 0 , for both i = 1 an d 2. Let K = { 1 , 2 } . Next, we will prove the statement of T heorem 2.1 in the cases S 2 = B 1 and S 2 = B 2 , respectively . Case I: Let S 2 = B 1 . Note that in this case, for any finite-length w ord of the form w = ( i 1 , . . . , i ℓ , i ℓ + 1 , . . . , i ℓ + m ) = ( ℓ -time z }| { 1 , . . . , 1 , m -time z }| { 2 , . . . , 2) ∈ K ℓ + m , there holds S( w ) : = S i 1 · · · S i ℓ S i ℓ + 1 · · · S i ℓ + m = S ℓ 1 B m 1 = λ m      a ( ℓ ) 11 0 · · · 0 a ( ℓ ) 21 0 · · · 0 . . . . . . . . . . . . a ( ℓ ) d 1 0 · · · 0      , 5 for any ℓ ≥ 1 and m ≥ 1. Since { S 1 , S 2 } is p eriodically switched stable, S( w ) is expon entially stable and so it holds from the classical Gel’fand formula that ρ (S( w )) = | λ m a ( ℓ ) 11 | < 1 for all words w = ( ℓ -time z }| { 1 , . . . , 1 , m -time z }| { 2 , . . . , 2), for all ℓ ≥ 1 and m ≥ 1. Let i · : N → K be an arb itrary switching sign al. If to any N ≥ 1 there is som e n ≥ N so that the in finite-length seque nce i · = ( i 1 , i 2 , . . . ) contain s at least one o f th e fo llowing tw o sub- words of finite-length n ( n -time z }| { 1 , . . . , 1) a nd ( n -time z }| { 2 , . . . , 2 ) , then it holds that | | | | S i 1 · · · S i n | | || ∗ → 0 as n → + ∞ . Hence, we only need to consider the following special case: i · = ( ℓ 1 -time z }| { 1 , . . . , 1 , m 1 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 1 , ℓ 2 -time z }| { 1 , . . . , 1 , m 2 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 2 , . . . ... , ℓ n -time z }| { 1 , . . . , 1 , m n -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w n , . . . ) where 1 ≤ ℓ n ≤ L and 1 ≤ m n ≤ M fo r all n ≥ 1, f or some two positive integers L ≥ 1 an d M ≥ 1. Theref ore, there exists a po siti ve constant γ = γ ( L , M ) < 1 such that | λ m n a ( ℓ n ) 11 | = ρ (S( w n )) ≤ γ ∀ n ≥ 1 . Notice here that for the gi ven special switching signal i · : N → K , γ is ind ependen t o f the ex- tremal norm | | | | · | | || ∗ of S used here. From the fact that lim sup n → + ∞ 1 n log || | | S i 1 · · · S i n | | || ∗ = lim su p n → + ∞ 1 J n log || | | S i 1 · · · S i J n | | || ∗ where J n = n X k = 1 ( ℓ k + m k ) = lim su p n → + ∞ 1 P n k = 1 ( ℓ k + m k ) log Y n k = 1 | λ m k a ( ℓ k ) 11 | ≤ 1 L + M log γ < 0 by [ 14 , Theorem 2.1] and the triangular ity o f S( w n ), it follows at once that | | | | S i 1 · · · S i n | | || ∗ → 0 as n → + ∞ . Since the switching signa l i · : N → K is arbitrary her e, this proves that { S 1 , S 2 } is absolutely asymptotically stable. Case (II): Let S 2 = B 2 . Noting that S ℓ 1 S 2 =      a ( ℓ ) 12 0 · · · 0 a ( ℓ ) 22 0 · · · 0 . . . . . . . . . . . . a ( ℓ ) d 2 0 · · · 0      ∀ ℓ ≥ 1 and S m 2 = 0 d × d ∀ m ≥ 2 , 6 we can prove, by an argument similar to that of the case (I) , that { S 1 , S 2 } is also absolutely asymptotically stable in this case. Now combinin g the cases (I) and (II) , we see that | | | | S i 1 · · · S i n | | || ∗ → 0 a s n → + ∞ for all switching signals i · : N → K . Th en, the statemen t o f T heorem 2.1 follows immediately from the Fenichel uniform ity th eorem proven in [ 21 ]. This completes the proof of Theorem 2.1 . If there is no the assumption o f rank 1 in the above Theore m 2.1 , then we can only guaran tee that S is exponen tially stable almo st surely in terms of some special probabilities from [ 15 , 13 ]. As a result of Theore m 2 .1 , we can obtain the following fin iteness property . Theorem 2.4. Let S = { S 1 , S 2 } ⊂ R d × d , where 2 ≤ d < + ∞ . If one of S 1 , S 2 is of rank 1 , then S has the spectral finiteness pr o perty; that is, one can found some finite n ≥ 1 such that ρ ( S ) = max n n p ρ (S i 1 · · · S i n ) : ( i 1 , . . . , i n ) ∈ K n o . Her e K = { 1 , 2 } . Pr oof. Th ere is no loss of g enerality in assumin g ρ ( S ) = 1, by n ormalization of S if n ecessary . Suppose, by contrad iction, that ρ ( A ) < 1 ∀ A ∈ S + . Then from Th eorem 2.1 , it follows that the switched dy namics induced b y S is ab solutely expo- nentially stable. Thu s ˆ ρ ( S ) < 1 from [ 1 ] for exam ple, and further ρ ( S ) < 1 from the B erger-W ang formu la [ 4 ]. It is a contradictio n to the assumption of ρ ( S ) = 1 . This thus ends the proof of Theorem 2.4 . As a con sequence of Theo rem 2.4 , we can conclude the following r esult, which m eans that stability is algo rithmically dec idable for every pair s of real d × d matrices S 1 , S 2 one o f which has rank 1. Corollary 2 .5. Denote Z + = { 0 , 1 , 2 , . . . } . F or every pa irs of real d × d matrices S 1 , S 2 with rank(S 2 ) = 1 , we have ρ ( S ) = m ax ℓ, m ∈ Z + ℓ + m q ρ (S ℓ 1 S m 2 ) . Mor e sp ecifically , we have • if ρ (S 2 ) = 0 , then ρ ( S ) = max  max ℓ ∈ N ℓ + 1 q ρ (S ℓ 1 S 2 ) , ρ (S 1 )  • if ρ (S 2 ) , 0 , then ρ ( S ) = m ax ℓ, m ∈ Z + ℓ + m q ρ (S ℓ 1 S m 2 ) 7 Pr oof. Without lo ss of generality , we may assume S 2 = B 1 : =  λ 0 1 × ( d − 1) 0 ( d − 1) × 1 0 ( d − 1) × ( d − 1)  or S 2 = B 2 : =        0 0 0 · · · 0 1 0 0 · · · 0 0 0 0 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 0        . Similar to the previous proof of Theor em 2.1 , the possible optima l sequences should have th e form i · = ( ℓ 1 -time z }| { 1 , . . . , 1 , m 1 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 1 , ℓ 2 -time z }| { 1 , . . . , 1 , m 2 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 2 , . . . ... , ℓ n -time z }| { 1 , . . . , 1 , m n -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w n , . . . ) , by noting that ρ (B ℓ 1 S( w 1 w 2 · · · w n )) = ρ (S( w ′ 1 w 2 · · · w n )) where w ′ 1 = ( ℓ 1 -time z }| { 1 , . . . , 1 , ( m 1 + ℓ )-time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ ) and ρ (B ℓ 2 S( w 1 w 2 · · · w n )) = 0 for all ℓ ≥ 1 and n ≥ 1. Denote i · ( n ) = w 1 w 2 · · · w n for any n ≥ 1. If S 2 = B 1 we then hav e ρ (S( i · ( n ))) = n Y k = 1 ρ (S 2 ) m k a ( ℓ k ) 11 which yields a maximu m when w 1 = w 2 = · · · = w n . In this case ρ (S( i · ( n ))) = ρ (S( w 1 ) n ) = ρ ( S ( w 1 )) n = ρ (S ℓ 1 S m 2 ) n . Now if we let α = sup ℓ, m ∈ N ℓ + m q ρ (S ℓ 1 S m 2 ) , then we have | i · ( n ) | p ρ (S( i · ( n ))) ≤ α. This gives ρ ( S ) ≤ α . On the other hand, we know that α = sup ℓ, m ∈ N ℓ + m q ρ (S ℓ 1 S m 2 ) ≤ ρ ( S ) 8 This leads to sup ℓ, m ∈ N ℓ + m p ρ (S ℓ 1 S m 2 ) = ρ ( S ) and so max ℓ, m ∈ N ℓ + m p ρ (S ℓ 1 S m 2 ) = ρ ( S ) fro m Theo- rem 2.4 . If S 2 = B 2 , the possible optimal sequence is giv en by i · = ( ℓ 1 -time z }| { 1 , . . . , 1 , m 1 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 1 , ℓ 2 -time z }| { 1 , . . . , 1 , m 2 -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w 2 , . . . ... , ℓ n -time z }| { 1 , . . . , 1 , m n -time z }| { 2 , . . . , 2 ✿✿✿✿✿✿✿✿✿✿✿✿✿ w n , . . . ) with m i ≡ 1. This correspon ds the pr evious case b y letting m 1 = m 2 = · · · = 1. Thus, the proof of Corollary 2.5 is completed . 3. Illustrated Examples In this section we p rovide several exam ples to illustrate our theoretical ou tcomes proved in Section 2 . W e he re p oint out that it is u nnecessary to tran sform the r ank-on e matrix S 2 to its Jordan cano nical form durin g practical calculation, since the cor respondin g optim al sequence is in variant un der similarity transfo rmation. Now , let u s car ry on the above an alysis o n the following examples. Example 1 (See [ 32 ]) . Let S = { S 1 , S 2 } , wher e S 1 =  1 0 1 1  , S 2 =  0 1 0 0  . Since S ℓ 1 =  1 0 ℓ 1  , we have S ℓ 1 S 2 =  1 1 0 ℓ  . Then ρ (S ℓ 1 S 2 ) = ℓ. Hence ρ ( S ) = max ℓ ∈ N ℓ + 1 √ ℓ = 5 √ 4 . This yields ρ ( S ) = 5 √ 4 and the correspon ding optimal sequenc e is S 4 1 S 2 . Example 2. S = ( S 1 =  1 1 √ 2 0 1  , S 2 = " 1 √ 3 2 − 1 − √ 3 2 #) . Notice that S ℓ 1 =  1 ℓ √ 2 0 1  , S m 2 = 1 − √ 3 2 ! m − 1 " 1 √ 3 2 − 1 − √ 3 2 # . and S ℓ 1 S m 2 = 1 − √ 3 2 ! m − 1  1 ℓ √ 2 0 1   1 − 1  h 1 √ 3 2 i . 9 Thus we have ρ (S ℓ 1 S m 2 ) = ℓ √ 2 + √ 3 2 − 1 ! 1 − √ 3 2 ! m − 1 . Hence ρ ( S ) = max ℓ, m ∈ N ℓ + m v u u t ℓ √ 2 + √ 3 2 − 1 ! 1 − √ 3 2 ! m − 1 = 6 s 5 √ 2 + √ 3 2 − 1 ≈ 1 . 22 6346 > max { ρ (S 1 ) , ρ (S 2 ) } , where the maximum is attained at ( ℓ, m ) = (5 , 1 ) with the optimal sequence S 5 1 S 2 . Example 3. S =  S 1 =  1 1 √ 2 0 1  , S 2 =  0 0 − 1 √ 2 1  . Notice that S ℓ 1 =  1 ℓ √ 2 0 1  , S m 2 = S 2 . and S ℓ 1 S m 2 =  ℓ √ 2 1  h − 1 √ 2 1 i . Thus we have ρ (S ℓ 1 S m 2 ) = | 1 − ℓ 2 | . Hence ρ ( S ) = m ax ℓ, m ∈ N ℓ + m r | 1 − ℓ 2 | = 11 √ 4 ≈ 1 . 1 3431 3 > m ax { ρ (S 1 ) , ρ (S 2 ) } , where the maximum is attained at ( ℓ, m ) = (10 , 1) with the optimal sequence S 10 1 S 2 . Example 4. S =        S 1 =     1 ε 0 0 0 1 ε 0 0 0 1 ε 0 0 0 1     , S 2 =     1 − 1 0 1 1 − 1 0 1 1 − 1 0 1 1 − 1 0 1            , wher e ε > 0 is a parameter . Notice that S ℓ 1 =     1 ℓ ε 1 2 ( ℓ − 1) ℓ ε 2 1 6 ( ℓ − 2)( ℓ − 1) ℓ ε 3 0 1 ℓε 1 2 ( ℓ − 1) ℓ ε 2 0 0 1 ℓε 0 0 0 1     , S m 2 = S 2 . 10 and S ℓ 1 S m 2 =     1 + ℓ ε + 1 2 ( ℓ − 1) ℓ ε 2 + 1 6 ( ℓ − 2)( ℓ − 1) ℓε 3 1 + ℓ ε + 1 2 ( ℓ − 1) ℓ ε 2 1 + ℓ ε 1      1 − 1 0 1  . Thus we have ρ (S ℓ 1 S m 2 ) = 1 6 ( ℓ − 2)( ℓ − 1) ℓ ε 3 + 1 . Hence ρ ( S ) = max ℓ, m ∈ N ℓ + m r 1 6 ( ℓ − 2)( ℓ − 1) ℓ ε 3 + 1 = max ℓ ≥ 3 ℓ + 1 r 1 6 ( ℓ − 2)( ℓ − 1) ℓ ε 3 + 1 = ℓ ε + 1 r 1 6 ( ℓ ε − 2 )( ℓ ε − 1 ) ℓ ε ε 3 + 1 , where the maximu m is assum ed to be ach iev ed at ℓ = ℓ ε . One can show that ℓ ε → ∞ as ε → 0. Numerical experiments in dicate that v alue o f ℓ ε increases very q uickly with respect to 1 ε . Thu s for any giv en integer L > 0, one always c an fin d a correspo nding con stant ε > 0, such that L p ρ (S i 1 S i 2 · · · S i L ) < ρ ( S ). This argument also can be easily sho wn by the fo llowing two-dimen sional exam ple. Example 5. S =  S 1 =  1 ǫ 0 1  , S 2 =  1 − 1 1 − 1  . Since S ℓ 1 =  1 ℓ ǫ 0 1  , S 2 2 = 0 we have S ℓ 1 S 2 =  1 + ℓ ǫ 1   1 − 1  . Then ρ (S ℓ 1 S 2 ) = ℓǫ . Hence ρ ( S ) = max ℓ ∈ N ℓ + 1 √ ℓǫ . Giv en any s pecified length L , let ǫ = 1 L + 1 , then ρ ( S ) = max ℓ ∈ N ℓ + 1 r ℓ L + 1 ≥ 1 > max 1 ≤ ℓ ≤ L ℓ + 1 r ℓ L + 1 , 11 where the last strictly inequality im plies that, for the ch osen ǫ = 1 L + 1 , the inten ded op timal sequence will never be fou nd within the length L . This special examp le represents the challenge ev en if we know the spectral finiteness prop erty holds. Therefo re, any algorithm s depend ing on the search of the length of optimal sequence will su ff er from a high compu tational cost. 4. Concluding remarks In this paper, we hav e proved that the spectral finiteness pro perty ho lds fo r every pairs of rea l d × d matrices S 1 , S 2 , if one of S 1 , S 2 has rank 1; see Theorem 1.3 . Under our context, S 1 and S 2 might be neith er symmetric, nor commu tativ e, and no r rational. In add ition, o ur a rgument do es not in volve any polytope norms. Recall that a matr ix A = [ a i j ] is called a binar y m atrix (r esp. sign-m atrix), p rovided that ev ery entries a i j belong to { 0 , 1 } (resp. {− 1 , 0 , 1 } ). In [ 32 , Theorem 4], R. Jungers and V . Blondel proved th at • Th e finiten ess prope rty ho lds fo r all sets of n onnegative rational squa re m atrices if and only it holds for all pairs of binary square matrices. • Th e finiten ess proper ty ho lds fo r all sets of rational sq uare matrices if and only it holds for all pairs of square sign-matrices. 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