PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS K.C. CHANG, KELLY PEARSON, AND TAN ZHANG

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS K.C. CHANG, KELLY PEARSON, AND TAN ZHANG Abstract. We geeralze the Perro Frobeus Theorem for oegatve matrces to the class of oegatve tesors. 1. Itroducto Perro Frobeus Theorem s a fudametal result for oegatve matrces. It has umerous applcatos, ot oly may braches of mathematcs, such as Markov chas, graph theory, game theory, ad umercal aalyss, but varous felds of scece ad techology, e.g. ecoomcs, operatoal research, ad recetly, page rak the teret, as well. Its fte dmesoal exteso s kow as the Kre Rutma Theorem for postve lear compact operators, whch has also bee wdely appled to Partal Dfferetal Equatos, Fxed Pot Theory, ad Fuctoal Aalyss. I late studes of umercal multlear algebra [7][4][1], egevalue problems for tesors have bee brought to specal atteto. I partcular, the Perro Frobeus Theorem for oegatve tesors s related to measurg hgher order coectvty lked objects [5] ad hypergraphs [6]. The purpose of ths paper s to exted Perro Frobeus Theorem to oegatve tesors. It s well kow that Perro Frobeus Theorem has the followg two forms: Theorem 1.1. (Weak Form) If A s a oegatve square matrx, the (1) r(a), the spectral radus of A, s a egevalue. (2) There exsts a oegatve vector x 0 =0such that (1.1) Ax 0 = r(a)x 0. We recall the followg defto of rreducblty of A: a square matrx A s sad to be reducble f t ca be placed to block upper-tragular form by smultaeous row/colum permutatos. A square matrx that s ot reducble s sad to be rreducble. Theorem 1.2. (Strog Form) If A s a rreducble oegatve square matrx, the (1) r(a) > 0 s a egevalue. (2) There exsts a oegatve vector x 0 > 0,.e. all compoets of x 0 are postve, such that Ax 0 = r(a)x 0. 1991 Mathematcs Subject Classfcato. Prmary 15A18, 15A69. Key words ad phrases. Numercal multlear algebra, hgher order tesor. 1

2 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG (3) (Uqueess) If λ s a egevalue wth a oegatve egevector, the λ = r(a). (4) r(a) s a smple egevalue of A. (5) If λ s a egevalue of A, the λ r(a). We shall exted these results to oegatve tesors. But frst, let us recall some deftos o tesors. A m-order -dmesoal tesor C s a set of m real etres (1.2) C =(c 1 m ), c 1 m R, 1 1,..., m. C s called oegatve (or respectvely postve) f c 1 m 0 (or respectvely c 1 m > 0). To a -vector x = (x 1,,x ), real or complex, we defe a -vector: (1.3) Cx m 1 := c 2 m x 2 x m. 1 Suppose Cx m 1 = 0, a par (λ, x) C (C \{0}) s called a egevalue ad a egevector, f they satsfy (1.4) Cx m 1 = λx [m 1], where x [m 1] =(x m 1 1,...,x m 1 ). Whe m s eve, ad C s symmetrc, ths was m 1 sg x1 troduced by Q [7]; whe m s odd, Lm [4] used (x1,...,x m 1 sg x ) o the rght-had sde stead, ad the oto has bee geeralzed Chag Pearso Zhag [1]. Ulke matrces, the egevalue problem for tesors are olear, amely, fdg otrval solutos of polyomal systems several varables. Ths feature eables us to employ dfferet methods geerazatos. The ma results of ths paper are stated as follows: Theorem 1.3. If A s a oegatve tesor of order m dmeso, the there exst λ 0 0 ad a oegatve vector x 0 =0such that (1.5) Ax m 1 = λ 0 x [m 1] 0. Theorem 1.4. If A s a rreducble oegatve tesor of order m dmeso, the the par (λ 0,x 0 ) equato (1.5) satsfy: (1) λ 0 > 0 s a egevalue. (2) x 0 > 0,.e. all compoets of x 0 are postve. (3) If λ s a egevalue wth oegatve egevector, the λ = λ 0. Moreover, the oegatve egevector s uque up to a multplcatve costat. (4) If λ s a egevalue of A, the λ λ 0. However, ulke matrces, such λ 0 s ot ecessarly a smple egevalue for tesors geeral. We shall preset a example to demostrate such dstcto. Furthermore, some addtoal codtos wll be mposed to esure the smplcty of the egevalue λ 0. I the paper of Lm [4], some of the above coclusos Theorem 1.4 were obtaed. However, we shall study ths problem more systematcally a more self-cotaed maer va a dfferet approach here. We orgaze our paper as follows: 2 s devoted to prove the ma theorems, except (4) of Theorem 1.4. I 3, we dscuss the smplcty of λ 0. I 4, we study a exteded Collatz s mmax Theorem, from whch asserto (4) of Theorem 1.4 wll

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 3 follow as a drect cosequece. I the last 5, varous extesos of the ma results wll be gve. 2. Proofs of the ma theorems Let X = R. It has a postve coe P = {(x 1,...,x ) X x 0, 1 }. The teror of P s deoted t P = {(x 1,...,x ) P x > 0, 1 }. A order s duced by P : x, y X, we defe x y f y x P, ad x<yf x y ad x = y. A m order tesor C s hece assocated wth a olear (m 1) homogeeous operator C : X X by Cx = Cx m 1, x X,.e., (2.1) C(tx) =t m 1 Cx, x X, t R 1. It s obvously see that f C s oegatve (or respectvely postve),.e, all etres are oegatve (or respectvely postve), the the assocate olear operator C : P P (or C : P \{0} t P ). Moreover, f C s oegatve, the (2.2) Cx Cy, x y, x, y P. Ad we are ow ready for the proof of Theorem 1.3: Proof. We reduce the problem to a fxed pot problem as follows. Let D = {(x 1,...,x ) X x 0, 1, =1 x = 1} be a closed covex set. Oe may assume Ax m 1 =0 x D. For otherwse, there exsts at least a x 0 D so that Ax m 1 0 = 0. Let λ 0 = 0, the (λ 0,x 0 ) s a soluto to (1.3), ad we are doe. The the followg map F : D D s well defed: (2.3) F (x) = (Ax m 1 ) 1 m 1, 1, j=1 (Axm 1 ) 1 m 1 j where (Ax m 1 ) s the th compoet of Ax m 1. F : D D s clearly cotuous. Accordg to the Brouwer s Fxed Pot Theorem, x 0 D such that F (x 0 )=x 0,.e. Ax m 1 0 = λ 0 x [m 1] 0, where m 1 (2.4) λ 0 = (Ax m 1 m 1. j=1 0 ) 1 j We ow tur to Theorem 1.4. If A s postve the we ca use smlar argumets used postve matrces to establsh coclusos (1) - (3) Theorem 1.4 based o Theorem 1.3. Our purpose the remag of ths secto s to troduce a codto o tesors whch les betwee postvty ad oegatvty to esure smlar results hold as Perro Frobeus Theorem for matrces. Defto 2.1. (Reducblty) A tesor C =(c 1 m ) of order m dmeso s called reducble, f there exsts a oempty proper dex subset I {1,...,} such that c 1 m =0, 1 I, 2,..., m / I. If C s ot reducble, the we call C rreducble.

4 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG Lemma 2.2. If a oegatve tesor C of order m dmeso s rreducble, the c 2 m > 0, 1. Proof. Suppose ot, the there exsts 0 so that c 2,..., m=1 0 2 m = 0. Sce C s oegatve, c 0 2 m =0 2,..., m. I partcular, f we let I = { 0 }, the c 1 2 m =0, 1 I ad 2,..., m / I, ths cotradcts rreducblty. We are ow ready for the proof of Theorem 1.4. Proof. 1 Frst, we prove x 0 t P. Note P \ t P = P = I Λ F I, where Λ s the set of all dex subsets I of {1,...,} ad F I = {(x 1,...,x ) P x =0 I,ad x j =0 j / I}. Suppose x 0 / t P, sce x 0 = 0, there must be a maxmal proper dex subset I Λ such that x 0 F I,.e. (x 0 ) =0 I ad (x 0 ) j > 0 j / I. Let δ = M{(x 0 ) j j/ I}, we the have δ>0. Sce x 0 s a egevector, Ax 0 F I,.e. (x 0 ) 2 (x 0 ) m =0, I. It follows δ m 1 2,..., m / I 2,..., m / I (x 0 ) 2 (x 0 ) m =0, I, hece we have = 0 I, 2,..., m / I,.e. A s reducble, a cotradcto. 2 Combg 1 ad Lemma 2.2, we have λ 0 > 0. 3 We ow prove the egevalue correspodg to the postve egevector s uque, amely, f (λ, x) ad (µ, y) R P are solutos of (1.5), the λ = µ. Accordg to 1 ad 2, such x, y t P ad λ, µ > 0. z t P ad w / P, we defe δ z (w) ={s R + z + sw P }, the δ z (w) > 0, z + tw P for 0 t δ z (w) ad z + tw P for t>δ z (w). Applyg these to (z,w) =(x, y), we have x ty P for 0 t δ z ( y). By defto ad (2.1), (2.2), (2.5) λx [m 1] = Ax m 1 δ x ( y) m 1 Ay m 1 = µδ x ( y) m 1 y [m 1], t follows x ( µ λ ) 1 m 1 δx ( y)y, thus µ λ. Lkewse, f we terchage x ad y, t follows y ( λ µ ) 1 m 1 δy ( x)x, ad thus λ µ. We have hece proved λ = µ. Therefore, the oly egevalue correspodg to the postve egevector s λ 0. 4 We prove the postve egevector s uque up to a multplcatve costat,.e. f x 0,x P \{0} satsfyg Ax m 1 0 = λ 0 x [m 1] 0 ad Ax m 1 = λ 0 x [m 1], the x = kx 0 for some costat k. It has bee kow that x 0 t P, by the defto of δ x0 ( x), we have x 0 tx P for 0 t δ x0 ( x) ad x 0 tx / P for t>δ x0 ( x). Ths mples x 0 t 0 x P, where t 0 = δ x0 ( x). So there exsts a oempty maxmal dex subset I {1,...,} such that x 0 t 0 x F I. If I = {1,...,},

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 5 the x 0 = t 0 x, ad we are doe. Otherwse, I s a oempty proper subset. There exst >0 ad δ>0 such that (x 0 ) δ, {1, 2,...}, 0 <t 0 x =(x 0 ), I, 0 < t 0x (x 0 ) < 1 / I, ad the I [(x 0 ) 2 (x 0 ) m t m 1 0 x 2 x m ]=λ 0 [(x 0 ) m 1 (t 0 x ) m 1 ]=0. We have ad It follows t m 1 0 x 2 x m (x 0 ) 2 (x 0 ) m 2,..., m, t m 1 0 x 2 x m (1 ) m 1 (x 0 ) 2 (x 0 ) m 2,..., m / I. δ m 1 (1 (1 ) m 1 ) 2,..., m / I 2,..., m / I [(x 0 ) 2 (x 0 ) m t m 1 0 x 2 x m ] [(x 0 ) 2 (x 0 ) m t m 1 0 x 2 x m ]=0 I, thus =0 I, 2,..., m / I,.e. A s reducble, a cotradcto. Remark: By the same argumet used 1 of the proof of Theorem 1.4, the followg mprovemet also holds: Assume A s a rreducble oegatve tesor. If x 0 P \{0} s a soluto of the equalty Ax m 1 λx [m 1], the x 0 t P. 3. The smplcty of the egevalue λ 0 For a matrx (.e. m = 2) A, a egevalue λ s called algebracally smple, f λ s a smple root of the characterstc polyomal det(a λi), ad s called geometrcally smple f dm Ker(A λi) = 1. We wll geeralze these otos to the tesor settg. Sce the operator A assocate wth a tesor A s olear but homogeeous, we ca defe the geometrc multplcty of a egevalue of A as follow: Defto 3.1. Let λ be a egevalue of (3.1) Ax m 1 = λx [m 1]. We say λ has geometrc multplcty q, f the maxmum umber of learly depedet egevectors correspodg to λ equals q. If q = 1, the λ s called geometrcally smple. It s worth otg the geometrc multplcty for a real egevalue λ of a real matrx A s depedet to the feld over the vector space beg real or complex,.e., dm R {x R (A λi)x =0} = dm C {z C (A λi)z =0}. Ths s due to the fact that f z = x + y R + R satsfes (A λi)z = 0, the both x, y Ker(A λi) R.

6 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG As to hgher order tesors, sce Ax m 1 s m 1 homogeeous, we stll have real geometrc multplcty complex geometrc multplcty, but ot equal geeral. Ths ca be see from the followg example: Example 3.2. Let m = 3 ad = 2. Cosder A =(a jk ) where a 111 = a 222 = 1, a 122 = a 211 = for 0 <<1, ad a jk = 0 for other (jk). The the egevalue problem becomes: x 2 1 + x 2 2 = λx 2 1 (3.2) x 2 1 + x 2 2 = λx 2 2. We have λ =1+ε, wth egevectors: u 1 = (1, 1) ad u 2 = (1, 1), ad λ =1 ε wth egevectors: u 3 = (1,), ad u 4 = (1, ). I ths example we see that real geometrc multplcty of λ = 1 + ε = complex geometrc multplcty = 2, ad real geometrc multplcty of λ = 1 ε s 0, ad complex geometrc multplcty s 2. The same example also shows the oegatve rreducble tesor A has a postve egevalue 1 + ε wth uque postve egevector (up to a multplcatve costat), whch s ot geometrcally smple ether R or C. Example 3.3. Let m =4,=2, A =(a jkl ) wth a 1222 = a 2111 = 1 ad a jkl =0 elsewhere. The after computato, we see there are two egevalues: λ = ±1, wth egevectors: (x, ±x), (x, ± exp 2π 4π 3 x), (x, ± exp 3 x). Therefore both λ = ±1 are all real geometrcally smple, but wth complex geometrcal multplcty 3. I the followg, we shall seek a suffcet codto to esure the real geometrc smplcty of λ 0. I case m s odd, there are two dfferet types of egevalue problems, whch mpose the same costrats o P : (1) Ax m 1 = λ(x m 1 1,...,x m 1 ), (2) Ax m 1 = λ(sg x 1 x m 1 1,...,sg x x m 1 ). Theorem 3.4. Let m be odd, ad let A be a rreducble oegatve tesor of order m dmeso. If Ax m 1 s varat uder ay oe of the trasformatos: (x 1,..., x ) (±x 1,..., ±x ), except the detty ad ts reflecto, the λ 0 s ot geometrcally smple for problem (1). If all terms Ax m 1 are moomals of x 2 1,...,x 2,.e. a 1 2 m =0oly f the umbers of dces appearg { 2,..., m } are all eve, 1, the λ 0 s real geometrcally smple for problem (2). Proof. (1) Let T be the trasformato, to whch Ax m 1 s varat uder. By assumpto, f x 0 =(x 0 1,...,x 0 ) t P s a soluto of (1), the Tx 0 s also a soluto of (1) correspodg to the same egevalue λ 0, so λ 0 s ot geometrcally smple. (2) By the assumpto, Ax m 1 0, x R, whch mples all solutos of (2) must be P. Usg asserto (3) of Theorem 1.4, we see x = kx 0,.e. λ 0 s real geometrcally smple.

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 7 We ext exame the case whe m s eve. We troduce a codto o C to esure the assocated olear operator C s creasg,.e. (3.3) x y Cx Cy. Comparg wth (2.2), there s o restrcto: x, y P (3.3). Defto 3.5. (Codto (M)) A tesor C = c 1 2 m of order m>2 dmeso s sad to satsfy Codto (M), f there exsts a oegatve matrx D =(d j ) such that c 1 2 m = d 1 2 δ 2 m, where δ 2 m s the Kroecker delta. Remark: For m = 2, Codto (M) s trval, hece s superfluous. I fact, f m s eve, Codto (M) o C mples (Cx m 1 ) =(m 1) d j x m 2 j 0, j, x j ad the Cx Cy, x y, x, y R. We ow state ad prove the followg: Theorem 3.6. Let m be eve, ad let A be a rreducble oegatve tesor. If A satsfes Codto (M), the the egevalue λ 0 for oegatve egevector s real geometrcally smple. Remark: To the specal problem, t ca, by settg y = x [m 1], be reduced to the problem for matrces, hece becomes a drect cosequece of Perro Frobeus Theorem. However, we preset the followg proof sce t wll be useful for more geeral problems, see 5. Proof. We follow 4 the proof of Theorem 1.4. We ote the oly dfferece s ow x R \{0} but ot P \{0}. We stll have t 0 = δ x0 ( x) such that x 0 tx P for 0 t t 0, ad x 0 tx / P for t>t 0. We wat to show x 0 = t 0 x. Suppose ot, oe has (x 0 ) δ>0, ad a oempty proper dex subset I such that t 0 x =(x 0 ) I ad t 0 x < (1 )(x 0 ) / I. It follows I δ m 1 (1 (1 ) m 1 ) j/ I j/ I j=1 a j j a j j [(x 0 ) m 1 j (t 0 x ) m 1 ] [(x 0 ) 2 (x 0 ) m t m 1 0 x 2 x m ]=0, thus a j j =0 j / I. Combg ths wth Codto (M), we obta a 1 2 m = 0 1 I, 2,..., m / I, whch cotradcts the rreducblty of A. Therefore, x 0 = t 0 x,.e. λ 0 s geometrcally smple as desred. We ext defe the algebrac smplcty of the egevalue of (3.1). We follow the approach descrbed Cox et al. [pp. 97 105] to defe the characterstc polyomal ψ A (λ) of A by ψ A (λ) := Res((Ax m 1 ) 1 λx m 1 1,...,(Ax m 1 ) λx m 1 )), where Res(P 1,...,P ) s the resultat of homogeeous polyomals P 1,...,P. For each A, such ψ A (λ) s uque up to a extraeous factor.

8 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG Defto 3.7. Let λ be a egevalue of (3.1). We say λ has algebrac multplcty p, f λ s a root of ψ A (λ) of multplcty p. Ad we call λ a algebracally smple egevalue, f p = 1. To the Example 3.2, we have kow that λ =1+ has geometrc multplcty 2 both real or complex felds. After computato we have 1 λ 0 0 ψ A (λ) = det 0 1 λ 0 0 1 λ 0 0 0 1 λ = (λ 1+) 2 (λ 1 ) 2, whch shows the egevalue λ 0 =1+ also has algebrac multplcty 2. By defto, we see complex geometrc multplcty algebrac multplcty, but ot equal geeral, ths ca be see the ext example. Example 3.8. Let m = 4 ad = 2. Cosder A =(a jkl ) where a 1111 = a 1112 = a 2122 = a 2222 = 1, ad a jkl = 0 for other (jkl). The the egevalue problem becomes: x 3 1 + x 2 1x 2 = λx 3 1 x 1 x 2 2 + x 3 2 = λx 3 2. We compute to see 1 λ 1 0 0 0 0 0 1 λ 1 0 0 0 ψ A (λ) = det 0 0 1 λ 1 0 0 0 0 1 1 λ 0 0 0 0 0 1 1 λ 0 0 0 0 0 1 1 λ = λ(λ 2)(λ 1) 4, whch shows the egevalues λ = 0, 2 are all algebracally ad geometrcally smple, wth egevectors u 1 = (1, 1), ad u 2 = (1, 1) respectvely, whle λ = 1 has algebrac multplcty 4. but has oly two learly depedet egevectors u 3 = (1, 0) ad u 4 = (0, 1), so ts geometrc multplcty s 2. 4. A Mmax Theorem The followg well kow [3] mmax theorem for rreducble oegatve matrces wll be exteded to rreducble oegatve tesors. Theorem 4.1. (Collatz) Assume that A s a rreducble oegatve matrx, the (4.1) M x t P Max x>0 (Ax) x = λ 0 = Max x t P M x>0 (Ax) x, where λ 0 s the uque postve egevalue correspodg to the postve egevector. I the remader of ths secto, we wll prove the followg

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 9 Theorem 4.2. Assume that A s a rreducble oegatve tesor of order m dmeso, the (4.2) M x t P Max x>0 (Ax m 1 ) x m 1 = λ 0 = Max x t P M x>0 (Ax m 1 ) x m 1, where λ 0 s the uque postve egevalue correspodg to the postve egevector. Before we proceed wth the proof of Theorem 4.2, we frst defe the followg two fuctos o P \{0}: µ (x) = M x>0 (Ax m 1 ) x m 1 ad µ (x) = Max x>0 (Ax m 1 ) x m 1. Clearly, µ (x) µ (x). Note µ (x) may be + o the boudary P \{0}. Sce both µ (x) ad µ (x) are postve 0-homogeeous fuctos, we ca restrct them o the compact set ={(x =(x 1,...,x ) P x =1}. Now, µ s cotuous ad bouded from above ad µ s cotuous o t P ad s bouded from below, there exst x,x such that =1 r := µ (x ) = Max x µ (x) = Max x P \{0} µ (x), r := µ (x ) = M x µ (x) = M x P \{0} µ (x). Let (λ 0,x 0 ) R + t P be the postve ege-par obtaed Theorem 1.4, we the have: Therefore, µ (x ) µ (x 0 )=λ 0 = µ (x 0 ) µ (x ). (4.3) r λ 0 r. We shall prove they are deed all equal. To do so, we modfy 3 the proof of Theorem 1.4 as follows: Lemma 4.3. Let A a rreducble oegatve tesor of order m dmeso. If (λ, x) ad (µ, y) R + (P \{0}) satsfy Ax m 1 = λx [m 1] ad Ay m 1 µy [m 1] (or respectvely Ay m 1 µy [m 1] ), the µ λ (or respectvely λ µ). Proof. We frst assume Ay m 1 µy [m 1]. Sce x t P, we have t 0 = δ x ( y) > 0 such that x ty P for 0 t t 0 ad x ty / P for t>t 0. It mples: λx [m 1] = Ax m 1 t m 1 0 Ay m 1 t m 1 0 µy [m 1], hece, x ( µ λ ) 1 m 1 t0 y, cosequetly, µ λ. Next we assume Ay m 1 µy [m 1]. After the remark of secto 2, we have y t P, ad f we terchage the roles of x ad y the prevous paragraph, the we have λ µ. Our asserto ow follows. We ow retur to the proof of Theorem 4.2:

10 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG Proof. After (4.3), t remas to show r λ 0 r. By the defto of µ (x), we have Ths meas Lkewse, r = µ (x ) = M x>0 Ax m 1 r x [m 1]. (Ax m 1 Ax m 1 r x [m 1]. Our desred equalty follows from Lemma 4.3. (x ) m 1 ). Sce µ s cotuous o ad s 0-homogeeous, we have Corollary 4.4. λ 0 = Max x P \{0} M x>0 (Ax m 1 ) x m 1. We close ths secto by provg asserto (4) of Theorem 1.4: Proof. Let z C \{0} be a soluto of Az m 1 = λz [m 1] for some λ C. We wsh to show λ λ 0. Let y = z ad set y =(y 1,...,y ). Clearly, y P \{0}. Oe has (Az m 1 ) = z 2 z m y 2 y m =(Ay m 1 ). Ths shows λ y m 1 = λ z m 1 = (Az m 1 ) (Ay m 1 ). Applyg Corollary 4.4, we have λ M y>0 (Ay m 1 ) y m 1 Max x P \θ M x>0 (Ax m 1 ) x m 1 = λ 0. 5. Some Extesos There are varous ways defg egevalues for tesors, e.g., there are H egevalue, Z egevalue, D egevalue etc. see [7], [8],[9],[4]. They are ufed [1]. I ths secto, we exted the above results to more geeral egevalue problems for tesors. Let A ad B be two m order dmesoal real tesors. Assume both Ax m 1 ad Bx m 1 are ot detcally zero. We say (λ, x) C (C \{0}) s a ege-par (or egevalue ad egevector) of A relatve to B, f the -system of equatos (5.1) (A λb)x m 1 =0 possesses a soluto. The problem (1.5), whch s called the H egevalue problem s the case, where B =(δ 1 2... m ), the ut tesor. We ext troduce a few more codtos o oegatve tesors.

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 11 Defto 5.1. (Quas-dagoal) A tesor C of order m dmeso s sad to be quas-dagoal, f for all oempty proper dex subset I {1,...,}, c 1, 2,... m =0 for 1 / I ad 2,..., m I. Example 5.2. For m = 2, C s quas-dagoal f ad oly f t s a dagoal matrx. Example 5.3. If C =(δ 1... m ), where δ 1 m s the Kroecker delta, the C s quas-dagoal. Lemma 5.4. If a oegatve tesor C of order m dmeso s quas-dagoal, the there exsts M > 0 such that for all oempty proper dex subset I {1,...,}, oe has Ce I Me I, where e I =(e I 1,...,e I ) wth e I 1, I = 0, / I. Proof. Let M = c 1,..., m=1 1 m. We verfy (Ce I ) =0 / I by computg (C(e I ) m 1 ) = c 2 m e I 2 e I m = c 2 m =0, / I, provded by that C s quas-dagoal. 2,..., m I Defto 5.5. (Codto (E)) A oegatve tesor C of order m dmeso s sad to satsfy Codto (E), f there exsts a homeomorphsm C : R R such that (1) C P = C P, ad (2) x, y P, x y mples C 1 x C 1 y. For C =(δ 1... m ), C s the detty operator, so Codto (E) s satsfed. Example 5.6. Let m be eve, ad D be a postve defte matrx. If C s a m order dmesoal tesor satsfyg: Cx m 1 = Dx(Dx, x) m 2 1, the C satsfes (1) the Codto (E). Ideed, C 1 y = D 1 y(y, D 1 y) m 2 2(m 1). Example 5.7. Let us cosder the followg example: let C k : P P be the olear operator: C k x = x [2k 1] x 2(r k), m =2r x [2k] x 2(r k) m =2r +1, where 1 k r. Ad let C k =(c 1,..., m ) be a m order dmesoal oegatve tesor correspodg to C k, for example, c2... m x 2...x m = (5.2) = α =r k x 2k 1 (x 2 1 +... + x 2 ) r k m =2r, x 2k (x 2 1 +... + x 2 ) r k m =2r +1 (r k)! α 1!...α! x2α1 1...x 2α x 2k 1, m =2r, x 2k m =2r +1.

12 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG The left had sde equals to Σ β =m 1 Σ (2,..., m) (1 β 1,..., β ) c 2... m x β1 1...xβ, where (1 β1,..., β ) meas j s repeated for β j tmes, 1 j, ad ( 2,..., m ) ( 2,..., m) meas there exsts a π S m 1, the m 1 permutatve group, such that π( 2,..., m )=( 2,..., m). (5.2) mples that 2k 1 m =2r β j =2α j + δ j 2k m =2r +1. Therefore there exsts a represetato C k of C k such that c,2,..., m = 0 oly f l 2 such that l =. Cosequetly, C k s quas-dagoal. Also, C k satsfes Codto (E). I fact, defe C k = x [2k 1] x 2(r k) m =2r x [2k] x 2(r k) Sg(x) m =2r +1, where we use the otato: x [α] Sg(x) =(x α 1 sg(x 1 ),..., x α sg(x )). The y [ 1 (5.3) C 1 2k 1 ] (Σ =1 k = y 2 2k 1 ) r k m 1 m =2r y [ 1 2k ] Sg(y)(Σ =1 y 1 k ) r k m m =2r +1. Obvously, Ck satsfes (1) ad (2) the defto of Codto (E). Remark: For B = C 1 the problem (5.1) correspods to Z egevalue, ad whe m s eve for B = C m t correspods to H egevalue. 2 We have the followg geeral result: Theorem 5.8. Suppose that A ad B are oegatve tesors, ad that B satsfes (1) Codto (E), the there exst λ 0 0 ad a oegatve vector x 0 =0, such that (5.4) Ax m 1 0 = λ 0 Bx m 1 0. If further, we assume that A s rreducble ad B satsfes Codto (E), ad s quas dagoal, the x 0 t P, λ 0 > 0 ad s the uque egevalue wth oegatve egevectors. I partcular, for B = C k, the oegatve egevector s uque up to a multplcatve costat. Proof. We are satsfed oly to sketch the proof, because t s parallel to that secto 2. Let A, B be the olear operators correspodg to A, B respectvely. For the exstece part, we defe ad F (x) = ( B 1 Ax)) Σ j=1 ( B 1 Ax) j, λ 0 = (Σ j=1( B 1 Ax) j ) m 1 replacg of (2.3) ad (2.4). The after argumet s the same wth the couter part secto 2.

PERRON FROBENIUS THEOREM FOR NONNEGATIVE TENSORS 13 Next we follow step 1 the proof of theorem 2 to prove x 0 t P by cotradcto. Suppose ot, the there exsts a maxmal proper dex subset I such that x 0 F I. From the equato: Ax m 1 0 = λ 0 Bx m 1 0, ad that B s quas dagoal, t follows Bx 0 F I ad the Ax 0 F I. The followg argumets are the same. I step 3, (2.5) s replaced by λbx = Ax µδ x ( y) m 1 By. Sce B 1 s order preservg P, ad s postvely 1 m 1 homogeeous, we have λ 1 1 m 1 x µ m 1 δx ( y)y. Therefore µ λ. Aga the rest part s the same. As to the uqueess of the postve egevector (up to a multplcatve costat), we reduce the problem by chagg varables. For x = 0, let x x 2(r k) m α 1, m =2r ξ = x x 2(r k) m α 2, m =2r +1, where α =2k whe m s eve, ad 2k + 1 whe m s odd. The problem s the reduced to: Aξ = λ 0 ξ [α]. We shall prove that the oegatve egevector x 0 s uque. It s proved by cotradcto. Suppose ot, there exst x, y P \{0} satsfyg Ax = λ 0 x ad Ay = λ 0 y. Let ξ,η are the mages of x, y uder the above trasformato. The by the argumet step 4 of the proof Theorem 1.4 there exsts t>0 such that ξ = tη. Ths mples x = t( x y ) 2(r k) m α 1 y, where α =2k or 2k + 1 f m =2r or 2r + 1 resp. Lastly, the mmax theorem 4 s also exteded: (Ax m 1 ) (Ax m 1 ) M x t P Max x>0 (C k x m 1 = λ 0 = Max x t P M x>0 ) (C k x m 1. ) The followg steps follow the last paragraph of 4. Corollary 5.9. Theorem 1.3 holds for D-egevalue problem. Theorem 1.4 holds for H egevalue problem ad Z egevalue problem. More geerally, For Bx = x [k] ϕ(x), where 1 k [ m 2 ] ad ϕ(x) s a postvely m k 1 homogeeous postve polyomal, Perro Frobeus theorem also holds. Refereces 1. Chag, K.C., Pearso, K., Zhag, T., O egevalue problems of real symmetrc tesors, Pekg Uversty preprt (2008). 2. Cox, D., Lttle, J., O Shea, D. Usg Algebrac Geometry, Sprger 1998. 3. Collatz, L., Eschlessugssatz fur de charakterstsche Zahle vo Matrze Math. Z. 48, (1946), 221-226. 4. Lm, L.H., Sgular values ad egevalues of tesors, A varatoal approach, Proc. 1st IEEE Iteratoal workshop o computatoal advaces of mult-tesor adaptve processg, Dec. 13-15, 2005, 129 132.

14 K.C. CHANG, KELLY PEARSON, AND TAN ZHANG 5. Lm, L. H., Multlear pagerak: measurg hgher order coectvty lked objects, The Iteret : Today ad Tomorrow, July 2005. 6. Dreas, P., Lm, L. H., A multlear spectral theory f hypergraphs ad expader hypergraphs, 2005. 7. Q, L., Egevalues of a real supersymmetrc tesor, J. Symbolc Computato 40 (2005) 1302 1324. 8. Q, L., Su, W., Wag, Y., Numercal multlear algebra ad ts applcatos, Frot. Math Cha, 2 (2007), 501-528. 9. Q, L., Wag, Y., Wu, Ed. X., D-egevalues of dffuso kurtoss tesors, to appear Joural of Computatoal Mathematcs ad Applcatos. LMAM, School of Math. Sc., Pekg Uv., Bejg 100871, Cha, kcchag@math.pku.edu.c Dept. Math & Stats., Murray State Uv., Murray, KY 42071, USA, kelly.pearsog@murraystate.edu Dept. Math & Stats., Murray State Uv., Murray, KY 42071, USA, ta.zhag@murraystate.edu