Cyclic and Diagonal Products on a Mat r ix

We unify the theory of cyclic and diagonal products of elements of matrices. We obtain some new results on diagonal similarity, diagonal equivalence, complete reducibility and total support. 1. I N T R O D U C T I O N Cyclic products of e lements of matrices have been considered by Fiedler and P tak [4]. Diagonal products were considered b y Sinkhorn and Knopp Elll. We a t t emp t to give a unified t r ea tmen t of bo th concepts, thereby obta in ing some new results. In Sec. 2 we introduce the equivalence relations c and s on the set Dn of all n × n matrices wi th elements in an integral domain D. Here A c B if all corresponding cyclic products for A and B are equal (2.6), and A s B if all corresponding diagonal products are equal (2.8). We then introduce the operators A -~A c and A ~ A s. We obta in A c from A by set t ing equal to 0 all elements tha t do not lie on a nonzero cyclic product, and similarly, we obta in A s using diagonal products. We int roduce a part ial 0 order ~ on D~. I t is not hard to show tha t A 8 = sup{P,~-I(PoA)c: aesop,} , where S¢nl is the symmetr ic group on {1 . . . . . n}. We prove considerably more. If A has a k × (n + 1 k) block of zeros, and m = max{k, n + 1 k}, This author's current address is IBM, 101BM54, Owego, N. Y., 13827. This work was completed while on educational leave of absence from IBM. The research was supported in part by NSF Grant GP-17815, and by the Mathematics Research Center. University of Wisconsin, under Contract No. DA-31124-ARO-D-462. © American Elsevier Publishing Company, Inc., 1973 302 G.M. ENGEL AND H. SCHNEIDER then there exist permutat ion matrices Q1 . . . . . Qm such that 0 = s u p { Q i l ( Q i A ) c : i = 1 . . . . , m } , Lemma 2.33. Further, the bound m is sharp, (2.35). If A ' = 0, it follows tha t there are permutat ion matrices Q1 . . . . . Q~ such that A s = s u p { Q i l ( Q i A ) c : i = 1 . . . . . n} (2.37), and the bound n is sharp (2.38). If A s # 0, then there is a permutat ion matr ix P for which A ~ = P I ( P A ) c (2.37). In Sec. 3, we present a unified theory of irreducible and fully indecomposable matrices. Using our definitions, we present proofs of some known results, e.g. (3.14). Other results in this section are intuit ively obvious, but have not been stated before, since they require the definitions of Sec. 2 for a formal s tatement, e.g. (3.15), (3.16). In See. 4 we find necessary and sufficient conditions for two matrices with elements in a field to be diagonally similar and to be diagonally equivalent. We prove the following result (4.1), which is closely related to Fiedler and P tak [4, Theorem 3.121. The following are equivalent: (1) A £ B and (2) A c and B ~ are diagonally similar. An analogue is proved in (4.11). The following are equivalent : (1) A s B and (2) there exist diagonal matrices D1 and D2 such tha t A s = D I B S D 2 and per D 1 D 2 = 1. We show, (4.5), tha t A is irreducible if and only if, for all B, A c B implies tha t A and B are diagonally similar; and we prove an analogous condition for full indecomposability, (4.12). Let a ~ 5 be a mult ipl icat ive mapping of the field, and let B t be the transpose of B. We then apply the results of this section to determine necessary and sufficient conditions for a matr ix to be diagonally similar to a matr ix B where B = B t, Secs. 4.14 and 4.19. Corollaries 4.20 and 4.22 generalize results of Par ter and Youngs [9] concerning necessary and sufficient conditions for a matr ix to be diagonally similar to a symmetr ic or skew-symmetric matrix. In Sec. 5 we show tha t a fully indecomposable matr ix A is diagonally equivalent to a (0, 1) matr ix if and only if all nonzero diagonal products equal a constant with an n th root in the field (5.2). When A is a nonnegat ive matrix, this result is due to Sinkhorn and Knopp [ l l j . Analogously, an irreducible matr ix A is diagonally similar to a (0, 1) matr ix if and only if all nonzero cyclic products equal 1 (5.8). In Sec. 6 we consider real and complex matrices, and present characterizations of full indecomposability, (6.10), and total support (6.13) : Let A >~ 0. Then A is total ly supported if and only if, for all B, tBI ~< A and ]per B t -per A imply there exist diagonal matrices D1, D 2 whose diagonal elements have absolute value 1, such that A = D 1 B D 2 . As Corollary 6.14, we obtain: CYCLIC AND D I A G O N A L P R O D U C T S 303 Le t A >~0. If I B I ~ A a n d Ipe rB[ = p e r A , t hen A* = D I B * D 2 where IOll = IDd = I . Th i s co ro l l a ry pa ra l l e l s Os t rowsk i [8, Z u s a t z zu Sa t z I I : L e t A be a n o n s i n g u l a r M m a t r i x . If B is a m a t r i x w i th Ibiil <~ aii a n d Ibijt >~ a . a n d Idet B I = de t A then B * = D1A*D2, where D1, D2 are d i agona l m a t r i c e s w i t h IDl] = ID2I = I .