- Email: [email protected]
Integrally normalizable matrices with respect to a given set
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.1 (1-9)
Linear Algebra and its Applications ••• (••••) •••–•••
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Integrally normalizable matrices with respect to a given set ✩ Sudipta Mallik a,∗ , Bryan L. Shader b a
Department of Mathematics & Statistics, Northern Arizona University, 805 S. Osborne Dr. PO Box: 5717, Flagstaff, AZ 86011, USA b Department of Mathematics, University of Wyoming, 1000 E University Avenue, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 10 November 2014 Accepted 2 July 2015 Available online xxxx Submitted by R. Brualdi MSC: 15B36 05C20 05C50 Keywords: Diagonal similarity Spanning tree Integer matrix
a b s t r a c t The n × n matrix A is integrally normalizable with respect to a prescribed subset M of {(i, j) : i, j = 1, 2, . . . , n and i = j} provided A is diagonally similar to an integer matrix each of whose entries in positions corresponding to M is equal to 1. In the case that the elements of M form the arc set of a spanning tree, the matrices that are integrally normalizable with respect to M have been characterized. This paper gives a characterization for general subsets M . In addition, necessary and sufficient conditions for each matrix with a given zero– nonzero pattern to be integrally normalizable with respect to an arbitrary subset M are given. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Let A be an n × n integer matrix and M be a subset of {(i, j) : i, j = 1, 2, . . . , n and i = j}. ✩
Dedicated to Hans Scheider whose results form the basis of our work.
* Corresponding author. E-mail addresses: [email protected] (S. Mallik), [email protected] (B.L. Shader). http://dx.doi.org/10.1016/j.laa.2015.07.001 0024-3795/© 2015 Elsevier Inc. All rights reserved.
JID:LAA 2
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.2 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
The matrix A is integrally normalizable with respect to M if there exists an n×n (rational) invertible diagonal matrix D such that DAD−1 is an integer matrix and (DAD−1 )i,j = 1 for all (i, j) in M . Integrally normalizable matrices were introduced and studied in [2] where the cardinality, |M |, of M was restricted to be n − 1. We abbreviate the phrase “A is integrally normalizable with respect to M ” to “A is INM ”. The digraph D (A) has vertex set V = {1, . . . , n} and arc set Γ(A) = {(i, j) : ai,j = 0}. A loop is an arc of the form (i, i). Since Γ(A) is a subset of {(i, j) : i, j = 1, 2, . . . , n}, we can view M as a subset of non-loop arcs of D (A). Underlying the directed graph D (A) is the undirected simple graph G(A) having vertex set V and an edge joining i and j in G(A) if and only if i = j, and (i, j) or (j, i) is in Γ(A). A digraph is weakly connected if its underlying undirected graph is connected. A subdigraph H of D (A) with arc set Γ is a spanning tree of D (A) provided |Γ| = n − 1 and the underlying graph of H is a spanning tree of G(A); that is, a connected graph on n vertices with no cycles. A sequence γ of arcs of the form (i1 , i2 ), (i2 , i3 ), . . . , (il , i+1 ) in a digraph is a walk from i1 to i+1 of length . A loop at vertex i corresponds to walk of length 1 from i to i. An empty sequence corresponds to a walk of length 0 from any vertex to itself. If i1 , i2 , . . . , i and i+1 are distinct, then γ is a directed path of length . If i+1 = i1 , then γ is a closed walk. If the vertices i1 , . . . , i in the closed walk (i1 , i2 ), (i2 , i3 ), . . . , (i , i+1 ) are distinct, then it is a directed cycle of length . If A = [ai,j ], then the weight of the walk γ in D (A) is wA (γ) = ai1 ,i2 ai2 i3 · · · ai i+1 . If the length of γ is 0, then wA (γ) = 1. The notions of walk, path, closed walk and cycles in G(A) are defined analogously. An undirected path (respectively, undirected cycle) is a collection of arcs which don’t form a direct path (respectively, cycle) whose underlying undirected graph is a path (respectively cycle). For a given subset S of {1, 2, . . . , n}, the set {1, 2, . . . , n}\S is denoted by S c . For subsets S and T of {1, 2, . . . , n}, A[S; T ] denotes the submatrix of A formed by the rows of A indexed by S and the columns of A indexed by T . The principal submatrix A[S; S] of A is denoted by A[S]. 2. Integrally normalizable matrices We begin with some terminology introduced in [2]. Let A = [aij ] be an n × n matrix, γ be a directed or undirected path, or a directed or undirected cycle of D (A) and (u, v) be an arc of γ. Let the edges of the underlying undirected graph of γ be {i1 , i2 }, {i2 , i3 }, . . . , {ik−1 , ik }. We say that (i, j) is a uv-forward arc of γ provided either both (u, v) and (i, j) are in {(i1 , i2 ), (i2 , i3 ), . . . , (ik−1 , ik )}
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.3 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–••• ⎡
0 ⎢ −3 ⎢ ⎢ A=⎢ 0 ⎢ ⎣ 0 0
2 0 0 0 0
0 4 0 −1 0
0 0 0 0 3
3
⎤ 3 0 ⎥ ⎥ ⎥ −2 ⎥ ⎥ 0 ⎦ 0 Fig. 1. A and D (A).
or both (u, v) and (i, j) are in {(i2 , i1 ), (i3 , i2 ), . . . , (ik , ik−1 )}. Thus (u, v) is always a uv-forward arc of γ. If (i, j) is not a uv-forward arc, then it is a F uv-reverse arc. Let γuv be the collection of uv-forward arcs of γ and set
F wA (γuv )=
aij .
F (i,j)∈γuv
R Similarly, let γuv be the collection of uv-reverse arcs of γ and set
R wA (γuv )=
aij .
R (i,j)∈γuv
As is customary, the product over the empty set is 1. When convenient, we omit the use F of uv, with the understanding that γ F denotes γuv for some chosen arc (u, v) of γ. Thus F F R γ is not uniquely defined, but the set {γ , γ } is independent of the choice of the arc (u, v) of γ. Example 2.1. Consider the matrix A and D (A) in Fig. 1. Then α = (3, 5), (5, 4), (4, 3) is a directed cycle, and β = (2, 1), (2, 3), (3, 5), (1, 5) F F and γ = (1, 2), (2, 3), (3, 5), (1, 5) are undirected cycles in D (A). Note that γ12 = γ23 = F R R R F γ35 = {(1, 2), (2, 3), (3, 5)} and γ12 = γ23 = γ35 = {(1, 5)}. Similarly γ15 = {(1, 5)} R and γ15 = {(1, 2), (2, 3), (3, 5)}. Although γ F and γ R are not unique, but {γ F , γ R } is independent of the choice of arc of γ. Lemma 2.2. (See [2, Lemma 2.1].) Let A be an n × n integer matrix and γ be a directed or undirected cycle in D (A). Let (u, v) be an arc of γ. Then F wA (γuv ) R ) wA (γuv
is invariant under diagonal similarity to A. Let A = [aij ] be an n × n matrix and M be a subset of non-loop arcs of D (A). We define CM (A) to be the set of all directed or undirected cycles γ in D (A) such that all the arcs (if any) of γ that are not in M are uv-forward arcs for some arc (u, v) of γ, i.e., R γuv ∩ (Γ(A) \ M ) = ∅.
JID:LAA 4
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.4 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
F R F For each γ ∈ CM (A), we define γM , γM and γM c as follows: F R F F γM = M ∩ γ F , γM := M ∩ γ R , and γM c = (Γ(A) \ M ) ∩ γ .
Note that we omit the use of uv in the preceding definitions with the understanding F R that γ F denotes γuv for some chosen arc (u, v) of γ for which γuv ∩(Γ(A) \ M ) = ∅. Thus F R F F R F γM , γM and γM c are not uniquely defined, but the set {γM , γM , γM c } is independent of the choice of the arc (u, v) of γ. Example 2.3. Consider the digraph D (A) in Example 2.1. Recall that α = (3, 5), (5, 4), (4, 3) is a directed cycle and β = (2, 1), (2, 3), (3, 5), (1, 5) and γ = (1, 2), (2, 3), (3, 5), (1, 5) are undirected cycles in D (A). Let M = {(1, 5)}. Then γ is in CM (A) and β does not belong to CM (A) because of the arcs (2, 1) and (2, 3). Moreover, CM (A) = {α, γ}. F R F Note that for the forward arc (1, 2), γM = ∅, γM = {(1, 5)} and γM = C R ∩ (Γ(A) \ M ) = {(1, 2), (2, 3), (3, 5)} = ∅, the arc (1, 5) is {(1, 2), (2, 3), (3, 5)}. Since γ15 not used as a forward arc of γ whenever γ is taken from CM (A). For each γ ∈ CM (A) and for each n × n matrix B = [bi,j ] with the same zero–nonzero pattern as A, we define F wB (γM )=
F {bi,j : (i, j) ∈ γM },
R {bi,j : (i, j) ∈ γM }, F F wB (γM {bi,j : (i, j) ∈ γM c) = c }. R wB (γM )=
R By definition, wA (γM c ) = 1. Hence,
wA (γ F ) wA (γ F ) = . R) R wA (γ ) wA (γM This, along with Lemma 2.2, imply the following result. Lemma 2.4. Let A be an n × n integer matrix, M be a subset of non-loop arcs of D (A) and B be an integer matrix that is diagonally similar to A. Then wB (γ F ) wA (γ F ) = R R) wA (γM ) wB (γM for each γ in CM (A). The following gives necessary conditions for an integer matrix A to be integrally normalizable with respect to a given M , and generalizes Theorem 2.4 of [2], which concerns the case that M is a spanning tree of D (A).
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.5 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
5
Theorem 2.5. Let A be an n × n integer matrix and M be a subset of non-loop arcs of D (A). If A is INM , then R ) | wA (γ F ) for each γ in CM (A), and (a) wA (γM R F (b) wA (γM ) = wA (γM ) if each arc of γ is in M .
Proof. Let A be INM . Then there is an integer matrix B = [bij ] which is diagonally similar to A and has bi,j = 1 for all (i, j) ∈ M . Consider γ in CM (A). By Lemma 2.4, wB (γ F ) wA (γ F ) = . R R) wA (γM ) wB (γM F R Since bi,j = 1 for all (i, j) ∈ M , we have wB (γM ) = 1 = wB (γM ) = 1, and wA (γ F ) = F wA (γM c ). Thus
wA (γ F ) F = wB (γM c ). R) wA (γM
(1)
F Since B is an integer matrix, wB (γM c ) is an integer and consequently (a) holds. If each arc of γ is in M , then (1) implies that (b) holds. 2
The following theorem, which is a re-phrasing of [2, Thm 2.5] into our terminology, characterizes the matrices A that are integrally normalizable with respect to a set M where the elements of M form the arc set of a spanning tree in D (A). In this case there is no directed or undirected cycle of D (A) having each of its arcs in M . Thus, Theorem 2.6 shows that the converse of Theorem 2.4 is true in the case that M is the arc set of a spanning tree of D (A). Note that in this case, if (u, v) is an arc of D (A) not in M , then the underlying graph of M ∪ {(u, v)} contains a unique cycle. Theorem 2.6. Let A be an n × n integer matrix and Γ be a subset of non-loop arcs of D (A) such that Γ forms the arc set of a spanning tree of D (A). Then A is INΓ if and only if for each arc (u, v) of D (A) that is not in Γ, the directed or undirected cycle γ consisting of arcs in Γ along with the arc (u, v) satisfies R F wA (γuv ) | wA (γuv ).
Let A be an n × n matrix and M be a subset of non-loop arcs of D (A). We define PM (i, j) to be the set of all directed and undirected paths γ = (i, j) from i to j for which there is an arc (u, v) in γ such that all arcs of γ not in M are uv-forward arcs in γ. The following theorem is the main result of this paper which gives necessary and sufficient conditions for an integer matrix to be integrally normalizable. Theorem 2.7. Let A be an n × n integer matrix and M be a subset of non-loop arcs of D (A). Then A is INM if and only if
JID:LAA 6
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.6 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
R (a) wA (γM ) | wA (γ F ) for each γ in CM (A) and F R (b) wA (γM ) = wA (γM ) for each cycle γ each of whose arcs lies in M .
Proof. Theorem 2.5 asserts that if A is INM , then (a) and (b) hold. We now prove the converse by induction on |M |. First consider the case that M is empty. Then each n × n integer matrix is INM . Thus, the base case of the induction is verified. Now assume that M is nonempty, say |M | = k + 1 with k ≥ 0, (a) and (b) hold for A, and whenever the analogs of (a) and (b) hold for an n × n matrix B and a subset N with |N | ≤ k, the matrix B is INN . Choose some (i, j) ∈ M . Let N = M \{(i, j)}. Note that each directed or undirected cycle in CN (A) is also in CM (A). Hence, the necessary and sufficient conditions ((a) and (b) in Theorem 2.7) for A to be INN are met, and we conclude that A is diagonally similar to an integer matrix B with br,s = 1 for all (r, s) ∈ N . By Lemma 2.2, the analogs of (a) and (b) hold for B. If bi,j = 1, then B is INM and we are done. Now suppose that bi,j = −1. Let H be the subgraph of G(A) consisting of the edges rs for which (r, s) ∈ M . Since the analog of (b) holds for B, ij is a bridge of H. Let D be the n × n diagonal matrix whose rth diagonal entry is −1 if there is a walk in H from r to i not using j, and is 1 otherwise. Then (D−1 BD)r,s = 1 for all (r, s) ∈ M , and we conclude that A is INM . Finally, suppose that |bi,j | > 1, and let p be a prime divisor of bi,j . Consider β ∈ PN (i, j). The arcs of β along with (i, j) form the undirected cycle γ which lies in CM (A) and has (i, j) as a reverse arc. By Lemma 2.4, wB (γ F ) wB (β) wA (γ F ) = = . R) R) bi,j wA (γM wB (γM
(2)
R Since wA (γM ) | wA (γ F ), bi,j | wB (β) for all β ∈ PN (i, j). Let S be the set of vertices x in D (B) such that there exists γ ∈ PN (i, x) such that wB (γ) is not multiple of p. Note that i ∈ S since there is a directed path of length 0 from i to i. Also note that j ∈ / S since wB (γ) is multiple of bi,j for all γ ∈ PN (i, j). Let M1 , . . . , Mt be the weakly connected components of D (A) and let V1 , . . ., Vt be the corresponding vertex sets. We shall show that for each V either V ⊆ S or V ⊆ S c . Suppose V ∩ S = ∅. For each v in V ∩ S, there is a shortest directed path in D (A) from i to v whose weight is not divisible by p. Among all such v, choose w to have the shortest path π whose weight is not divisible by p. Consider x ∈ V . Since M is weakly connected, there is a sequence β of arcs in M such that underlying graph of β is a path from w to x. The choice of π implies that the arcs of π along with the arcs in β form a path α in PN (i, x) and wB (α) = wB (π) is not multiple of p. Thus x ∈ S, and we conclude that each vertex of V is in S. Now we show that each entry of B[S; S c ] is a multiple of p. Consider an arc (, m) with ∈ S and m ∈ S c . Then there exists γ ∈ PN (i, ) such that p wB (γ). Since m ∈ S c ,
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.7 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
7
γ does not contain the vertex m. Hence, the arcs of γ along with the arc (, m) form a path in PN (i, m) whose weight equals wB (γ)b,m . Since m ∈ S c this weight is a multiple of p. Since p is a prime which does not divide wB (γ), we conclude that p divides b,m . Hence each entry of B[S; S c ] is a multiple of p. By applying a permutation similarity if necessary, we may assume without loss of generality that S = {1, 2, . . . , |S|}. Let D be the diagonal matrix with di = 1/p for i = 1, . . . , |S|. Let C = DBD−1 . Since C is obtained from B by multiplying its first |S| rows (respectively, columns) by 1/p (respectively, p), C is an integer matrix and Ci,j = d/p. Thus C = DBD−1 is an integer matrix with (i, j)-entry d/p. Note that entries of C corresponding to N are all 1 because for each Vi , either Vi ⊆ S or Vi ⊆ S c . wC (γ) wB (γ) = for all β ∈ PN (i, j). Now by repeated By (2) and Lemma 2.4, d d/p application of the above technique can find a matrix which is diagonally similar to B whose entries corresponding to N are all 1, and whose (i, j) entry is ±1. These cases have been previously handled. Therefore, we conclude that B, and hence A, is INM . 2 3. Integrally normalizable patterns A zero–nonzero pattern A is a matrix A = [αi,j ] where αi,j ∈ {0, ∗}. A matrix A = [ai,j ] is an integer realization of A if ai,j = 0 when αi,j = 0 and ai,j is a non-zero integer when αi,j = ∗. The set of all integer realizations of A is denoted by QZ (A). Let A be an n × n zero–nonzero pattern and M be a subset of {(i, j) : i, j = 1, 2, . . . , n and i = j}. The zero–nonzero pattern A is integrally normalizable with respect to M if A is integrally normalizable with respect to M for each A ∈ QZ (A). Necessary and sufficient conditions for an n × n zero–nonzero pattern A with a spanning tree T in D (A) to be integrally normalizable with respect to the arcs of T are given in [2, Theorem 3.1]. In this section we give necessary and sufficient conditions for a zero–nonzero pattern A to be integrally normalizable with respect to a given arc set M of D (A), not just M corresponding to a spanning tree of D (A). The digraph D (A) = (V, Γ(A)) has vertex set V = {1, . . . , n} and arc set Γ(A) = {(i, j) : αi,j = ∗}. Since Γ(A) is a subset of {(i, j) : i, j = 1, 2, . . . , n}, we can view M as a subset of the non-loop arcs of D (A). We define CM (A) analogously, i.e., CM (A) = CM (A) for all A ∈ QZ (A). The following theorem gives a necessary condition for a zero–nonzero pattern to be integrally normalizable. Theorem 3.1. Let A be an n × n zero–nonzero pattern and M be a nonempty, non-loop arc set of Γ(A). If A is integrally normalizable with respect to M , then Γ(M ) is a forest.
JID:LAA 8
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.8 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
Proof. Assume A is integrally normalizable with respect to M . Then for each A ∈ QZ (A), there is B ∈ QZ (A) such that B = [bi,j ] is diagonally similar to A and bi,j = 1 for all (i, j) ∈ M . Suppose that Γ(M ) contains a cycle γ. Then by Lemma 2.4, for each A ∈ QZ (A), wB (γ F ) wA (γ F ) = = 1. R) R) wA (γM wB (γM Since this is not possible for all A ∈ QZ (A), Γ(M ) contains no cycles. Thus Γ(M ) is a forest. 2 The following theorem gives necessary and sufficient conditions for a zero–nonzero pattern to be integrally normalizable. Theorem 3.2. Let A be an n × n zero–nonzero pattern and M be a non-loop arc set of Γ(A). Then A is integrally normalizable with respect to M if and only if Γ(M ) is a forest and each γ in CM (A) is a directed cycle. Proof. First suppose that A is integrally normalizable with respect to M . By TheoR ) | wA (γ F ) rem 3.1, Γ(M ) is a forest. Let γ be a cycle in CM (A). By Theorem 2.5, wA (γM R = ∅, we can construct a matrix in A ∈ QZ (A) by assigning for each A ∈ QZ (A). If γM 1 to entries corresponding to γ F and integers greater than 1 to entries corresponding to R R R γM . Then wA (γM ) does not divide 1 = wA (γ F ). Therefore γM = ∅ and consequently γ is a directed cycle in CM (A). Conversely suppose that Γ(M ) is a forest and each γ in CM (A) is a directed cycle. Let A be in QZ (A). Since Γ(M ) is a forest, there is no cycle γ in CM (A) = CM (A) each of whose arcs lies in M . Thus (b) of Theorem 2.7 holds for A. Let γ be in CM (A) = CM (A). R R Then γ is a directed cycle and γM = ∅. Consequently 1 = wA (γM ) divides wA (γ F ). Thus by Theorem 2.7, A is INM . Since this is true for all A in QZ (A), A is integrally normalizable with respect to M . 2 4. Rational matrices whose cycles have integer weights In this section we present a result that is similar to a result of G. Engel and Hans Schneider regarding diagonal similarity. In addition the proof is similar to that of Theorem 2.7. In particular in [1], Engel and Schneider prove that the n × n complex matrix A is diagonally similar to a (0, 1)-matrix if and only if the weight of each cycle in D (A) is 1. Here we give necessary and sufficient conditions for a given rational matrix to be similar to an integer matrix. Theorem 4.1. Let A = [aij ] be a rational matrix of order n. Then A is diagonally similar to an integer matrix if and only if wA (γ) is an integer for each directed cycle in D (A).
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.9 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
9
Proof. First suppose that A is diagonally similar to an integer matrix B. Let γ be a directed cycle of D (A). The analog of Lemma 2.2 for rationals, implies wA (γ) = wB (γ), and hence wA (γ) is an integer. We prove the converse by induction on the number of entries of A that are non-integer. If each entry of A is an integer, then we are done. Assume that wA (γ) is an integer for each directed cycle in D (A), at least one entry of A is not an integer and the result holds for all such matrices with fewer non-integer entries. Let aij be a non-integer entry of A. Consider the matrix B obtained from A by replacing aij by 0. Then each directed cycle in D (B) is a directed cycle in D (A), and hence has an integer weight. Moreover, by induction B is diagonally similar to an integer matrix. Thus, A is diagonally similar to a matrix each of whose entries other than the (i, j)-entry is an integer. Without loss of generality, we may assume each entry of A other than the (i, j)-entry is an integer. Let r and s be relatively prime integers such that aij = r/s. Since |s| > 1, there exists a prime divisor p of s. Take S to be the set of all vertices v such that there exists a directed path from j to v in D (A) whose weight is not divisible by p. Then j ∈ S. Also, if there exists a directed path α from j to i, then α along with the arc (i, j) forms a directed cycle γ whose weight is an integer. As wA (γ) = wA (α)(r/s), we conclude that p / S. Now consider an arc of the form (u, v) where divides wA (α) for each such α. Thus, i ∈ u ∈ S and v ∈ / S. As u ∈ S, there exists a directed path β from j to u whose weight is an integer not divisible by p. The directed path β along with the arc (u, v) forms a directed path from j to v whose weight is an integer. As v ∈ / S the weight of this path must be divisible by the prime p. It follows that p divides auv . Moreover, each entry of A[S, S c ] is an integer divisible by p. Now consider the diagonal matrix D = diag(d1 , . . . , dn ) with dk = p if k ∈ S and dk = 1 otherwise. Then D−1 AD is matrix all of whose entries are integer except possibly the (i, j)-entry, and each of directed cycles in its digraph has integer weight. Note the (i, j) entry of D−1 AD is r/(s/p). If (s/p) = ±1, we are done. Otherwise, we replace A by D−1 AD and repeat the above process. As the denominators of the (i, j)-entries of the matrices obtained decrease in magnitude, we eventually end with an integer matrix that is diagonally similar to A. 2 References [1] G.M. Engel, H. Schneider, Cyclic and diagonal products on a matrix, Linear Algebra Appl. 7 (1973) 301–335. [2] C. Garnett, D.D. Olesky, B.L. Shader, P. van den Driessche, Integrally normalizable matrices and zero–nonzero patterns, Linear Algebra Appl. (2014) 132–153.
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.1 (1-9)
Linear Algebra and its Applications ••• (••••) •••–•••
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Integrally normalizable matrices with respect to a given set ✩ Sudipta Mallik a,∗ , Bryan L. Shader b a
Department of Mathematics & Statistics, Northern Arizona University, 805 S. Osborne Dr. PO Box: 5717, Flagstaff, AZ 86011, USA b Department of Mathematics, University of Wyoming, 1000 E University Avenue, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 10 November 2014 Accepted 2 July 2015 Available online xxxx Submitted by R. Brualdi MSC: 15B36 05C20 05C50 Keywords: Diagonal similarity Spanning tree Integer matrix
a b s t r a c t The n × n matrix A is integrally normalizable with respect to a prescribed subset M of {(i, j) : i, j = 1, 2, . . . , n and i = j} provided A is diagonally similar to an integer matrix each of whose entries in positions corresponding to M is equal to 1. In the case that the elements of M form the arc set of a spanning tree, the matrices that are integrally normalizable with respect to M have been characterized. This paper gives a characterization for general subsets M . In addition, necessary and sufficient conditions for each matrix with a given zero– nonzero pattern to be integrally normalizable with respect to an arbitrary subset M are given. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Let A be an n × n integer matrix and M be a subset of {(i, j) : i, j = 1, 2, . . . , n and i = j}. ✩
Dedicated to Hans Scheider whose results form the basis of our work.
* Corresponding author. E-mail addresses: [email protected] (S. Mallik), [email protected] (B.L. Shader). http://dx.doi.org/10.1016/j.laa.2015.07.001 0024-3795/© 2015 Elsevier Inc. All rights reserved.
JID:LAA 2
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.2 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
The matrix A is integrally normalizable with respect to M if there exists an n×n (rational) invertible diagonal matrix D such that DAD−1 is an integer matrix and (DAD−1 )i,j = 1 for all (i, j) in M . Integrally normalizable matrices were introduced and studied in [2] where the cardinality, |M |, of M was restricted to be n − 1. We abbreviate the phrase “A is integrally normalizable with respect to M ” to “A is INM ”. The digraph D (A) has vertex set V = {1, . . . , n} and arc set Γ(A) = {(i, j) : ai,j = 0}. A loop is an arc of the form (i, i). Since Γ(A) is a subset of {(i, j) : i, j = 1, 2, . . . , n}, we can view M as a subset of non-loop arcs of D (A). Underlying the directed graph D (A) is the undirected simple graph G(A) having vertex set V and an edge joining i and j in G(A) if and only if i = j, and (i, j) or (j, i) is in Γ(A). A digraph is weakly connected if its underlying undirected graph is connected. A subdigraph H of D (A) with arc set Γ is a spanning tree of D (A) provided |Γ| = n − 1 and the underlying graph of H is a spanning tree of G(A); that is, a connected graph on n vertices with no cycles. A sequence γ of arcs of the form (i1 , i2 ), (i2 , i3 ), . . . , (il , i+1 ) in a digraph is a walk from i1 to i+1 of length . A loop at vertex i corresponds to walk of length 1 from i to i. An empty sequence corresponds to a walk of length 0 from any vertex to itself. If i1 , i2 , . . . , i and i+1 are distinct, then γ is a directed path of length . If i+1 = i1 , then γ is a closed walk. If the vertices i1 , . . . , i in the closed walk (i1 , i2 ), (i2 , i3 ), . . . , (i , i+1 ) are distinct, then it is a directed cycle of length . If A = [ai,j ], then the weight of the walk γ in D (A) is wA (γ) = ai1 ,i2 ai2 i3 · · · ai i+1 . If the length of γ is 0, then wA (γ) = 1. The notions of walk, path, closed walk and cycles in G(A) are defined analogously. An undirected path (respectively, undirected cycle) is a collection of arcs which don’t form a direct path (respectively, cycle) whose underlying undirected graph is a path (respectively cycle). For a given subset S of {1, 2, . . . , n}, the set {1, 2, . . . , n}\S is denoted by S c . For subsets S and T of {1, 2, . . . , n}, A[S; T ] denotes the submatrix of A formed by the rows of A indexed by S and the columns of A indexed by T . The principal submatrix A[S; S] of A is denoted by A[S]. 2. Integrally normalizable matrices We begin with some terminology introduced in [2]. Let A = [aij ] be an n × n matrix, γ be a directed or undirected path, or a directed or undirected cycle of D (A) and (u, v) be an arc of γ. Let the edges of the underlying undirected graph of γ be {i1 , i2 }, {i2 , i3 }, . . . , {ik−1 , ik }. We say that (i, j) is a uv-forward arc of γ provided either both (u, v) and (i, j) are in {(i1 , i2 ), (i2 , i3 ), . . . , (ik−1 , ik )}
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.3 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–••• ⎡
0 ⎢ −3 ⎢ ⎢ A=⎢ 0 ⎢ ⎣ 0 0
2 0 0 0 0
0 4 0 −1 0
0 0 0 0 3
3
⎤ 3 0 ⎥ ⎥ ⎥ −2 ⎥ ⎥ 0 ⎦ 0 Fig. 1. A and D (A).
or both (u, v) and (i, j) are in {(i2 , i1 ), (i3 , i2 ), . . . , (ik , ik−1 )}. Thus (u, v) is always a uv-forward arc of γ. If (i, j) is not a uv-forward arc, then it is a F uv-reverse arc. Let γuv be the collection of uv-forward arcs of γ and set
F wA (γuv )=
aij .
F (i,j)∈γuv
R Similarly, let γuv be the collection of uv-reverse arcs of γ and set
R wA (γuv )=
aij .
R (i,j)∈γuv
As is customary, the product over the empty set is 1. When convenient, we omit the use F of uv, with the understanding that γ F denotes γuv for some chosen arc (u, v) of γ. Thus F F R γ is not uniquely defined, but the set {γ , γ } is independent of the choice of the arc (u, v) of γ. Example 2.1. Consider the matrix A and D (A) in Fig. 1. Then α = (3, 5), (5, 4), (4, 3) is a directed cycle, and β = (2, 1), (2, 3), (3, 5), (1, 5) F F and γ = (1, 2), (2, 3), (3, 5), (1, 5) are undirected cycles in D (A). Note that γ12 = γ23 = F R R R F γ35 = {(1, 2), (2, 3), (3, 5)} and γ12 = γ23 = γ35 = {(1, 5)}. Similarly γ15 = {(1, 5)} R and γ15 = {(1, 2), (2, 3), (3, 5)}. Although γ F and γ R are not unique, but {γ F , γ R } is independent of the choice of arc of γ. Lemma 2.2. (See [2, Lemma 2.1].) Let A be an n × n integer matrix and γ be a directed or undirected cycle in D (A). Let (u, v) be an arc of γ. Then F wA (γuv ) R ) wA (γuv
is invariant under diagonal similarity to A. Let A = [aij ] be an n × n matrix and M be a subset of non-loop arcs of D (A). We define CM (A) to be the set of all directed or undirected cycles γ in D (A) such that all the arcs (if any) of γ that are not in M are uv-forward arcs for some arc (u, v) of γ, i.e., R γuv ∩ (Γ(A) \ M ) = ∅.
JID:LAA 4
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.4 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
F R F For each γ ∈ CM (A), we define γM , γM and γM c as follows: F R F F γM = M ∩ γ F , γM := M ∩ γ R , and γM c = (Γ(A) \ M ) ∩ γ .
Note that we omit the use of uv in the preceding definitions with the understanding F R that γ F denotes γuv for some chosen arc (u, v) of γ for which γuv ∩(Γ(A) \ M ) = ∅. Thus F R F F R F γM , γM and γM c are not uniquely defined, but the set {γM , γM , γM c } is independent of the choice of the arc (u, v) of γ. Example 2.3. Consider the digraph D (A) in Example 2.1. Recall that α = (3, 5), (5, 4), (4, 3) is a directed cycle and β = (2, 1), (2, 3), (3, 5), (1, 5) and γ = (1, 2), (2, 3), (3, 5), (1, 5) are undirected cycles in D (A). Let M = {(1, 5)}. Then γ is in CM (A) and β does not belong to CM (A) because of the arcs (2, 1) and (2, 3). Moreover, CM (A) = {α, γ}. F R F Note that for the forward arc (1, 2), γM = ∅, γM = {(1, 5)} and γM = C R ∩ (Γ(A) \ M ) = {(1, 2), (2, 3), (3, 5)} = ∅, the arc (1, 5) is {(1, 2), (2, 3), (3, 5)}. Since γ15 not used as a forward arc of γ whenever γ is taken from CM (A). For each γ ∈ CM (A) and for each n × n matrix B = [bi,j ] with the same zero–nonzero pattern as A, we define F wB (γM )=
F {bi,j : (i, j) ∈ γM },
R {bi,j : (i, j) ∈ γM }, F F wB (γM {bi,j : (i, j) ∈ γM c) = c }. R wB (γM )=
R By definition, wA (γM c ) = 1. Hence,
wA (γ F ) wA (γ F ) = . R) R wA (γ ) wA (γM This, along with Lemma 2.2, imply the following result. Lemma 2.4. Let A be an n × n integer matrix, M be a subset of non-loop arcs of D (A) and B be an integer matrix that is diagonally similar to A. Then wB (γ F ) wA (γ F ) = R R) wA (γM ) wB (γM for each γ in CM (A). The following gives necessary conditions for an integer matrix A to be integrally normalizable with respect to a given M , and generalizes Theorem 2.4 of [2], which concerns the case that M is a spanning tree of D (A).
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.5 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
5
Theorem 2.5. Let A be an n × n integer matrix and M be a subset of non-loop arcs of D (A). If A is INM , then R ) | wA (γ F ) for each γ in CM (A), and (a) wA (γM R F (b) wA (γM ) = wA (γM ) if each arc of γ is in M .
Proof. Let A be INM . Then there is an integer matrix B = [bij ] which is diagonally similar to A and has bi,j = 1 for all (i, j) ∈ M . Consider γ in CM (A). By Lemma 2.4, wB (γ F ) wA (γ F ) = . R R) wA (γM ) wB (γM F R Since bi,j = 1 for all (i, j) ∈ M , we have wB (γM ) = 1 = wB (γM ) = 1, and wA (γ F ) = F wA (γM c ). Thus
wA (γ F ) F = wB (γM c ). R) wA (γM
(1)
F Since B is an integer matrix, wB (γM c ) is an integer and consequently (a) holds. If each arc of γ is in M , then (1) implies that (b) holds. 2
The following theorem, which is a re-phrasing of [2, Thm 2.5] into our terminology, characterizes the matrices A that are integrally normalizable with respect to a set M where the elements of M form the arc set of a spanning tree in D (A). In this case there is no directed or undirected cycle of D (A) having each of its arcs in M . Thus, Theorem 2.6 shows that the converse of Theorem 2.4 is true in the case that M is the arc set of a spanning tree of D (A). Note that in this case, if (u, v) is an arc of D (A) not in M , then the underlying graph of M ∪ {(u, v)} contains a unique cycle. Theorem 2.6. Let A be an n × n integer matrix and Γ be a subset of non-loop arcs of D (A) such that Γ forms the arc set of a spanning tree of D (A). Then A is INΓ if and only if for each arc (u, v) of D (A) that is not in Γ, the directed or undirected cycle γ consisting of arcs in Γ along with the arc (u, v) satisfies R F wA (γuv ) | wA (γuv ).
Let A be an n × n matrix and M be a subset of non-loop arcs of D (A). We define PM (i, j) to be the set of all directed and undirected paths γ = (i, j) from i to j for which there is an arc (u, v) in γ such that all arcs of γ not in M are uv-forward arcs in γ. The following theorem is the main result of this paper which gives necessary and sufficient conditions for an integer matrix to be integrally normalizable. Theorem 2.7. Let A be an n × n integer matrix and M be a subset of non-loop arcs of D (A). Then A is INM if and only if
JID:LAA 6
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.6 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
R (a) wA (γM ) | wA (γ F ) for each γ in CM (A) and F R (b) wA (γM ) = wA (γM ) for each cycle γ each of whose arcs lies in M .
Proof. Theorem 2.5 asserts that if A is INM , then (a) and (b) hold. We now prove the converse by induction on |M |. First consider the case that M is empty. Then each n × n integer matrix is INM . Thus, the base case of the induction is verified. Now assume that M is nonempty, say |M | = k + 1 with k ≥ 0, (a) and (b) hold for A, and whenever the analogs of (a) and (b) hold for an n × n matrix B and a subset N with |N | ≤ k, the matrix B is INN . Choose some (i, j) ∈ M . Let N = M \{(i, j)}. Note that each directed or undirected cycle in CN (A) is also in CM (A). Hence, the necessary and sufficient conditions ((a) and (b) in Theorem 2.7) for A to be INN are met, and we conclude that A is diagonally similar to an integer matrix B with br,s = 1 for all (r, s) ∈ N . By Lemma 2.2, the analogs of (a) and (b) hold for B. If bi,j = 1, then B is INM and we are done. Now suppose that bi,j = −1. Let H be the subgraph of G(A) consisting of the edges rs for which (r, s) ∈ M . Since the analog of (b) holds for B, ij is a bridge of H. Let D be the n × n diagonal matrix whose rth diagonal entry is −1 if there is a walk in H from r to i not using j, and is 1 otherwise. Then (D−1 BD)r,s = 1 for all (r, s) ∈ M , and we conclude that A is INM . Finally, suppose that |bi,j | > 1, and let p be a prime divisor of bi,j . Consider β ∈ PN (i, j). The arcs of β along with (i, j) form the undirected cycle γ which lies in CM (A) and has (i, j) as a reverse arc. By Lemma 2.4, wB (γ F ) wB (β) wA (γ F ) = = . R) R) bi,j wA (γM wB (γM
(2)
R Since wA (γM ) | wA (γ F ), bi,j | wB (β) for all β ∈ PN (i, j). Let S be the set of vertices x in D (B) such that there exists γ ∈ PN (i, x) such that wB (γ) is not multiple of p. Note that i ∈ S since there is a directed path of length 0 from i to i. Also note that j ∈ / S since wB (γ) is multiple of bi,j for all γ ∈ PN (i, j). Let M1 , . . . , Mt be the weakly connected components of D (A) and let V1 , . . ., Vt be the corresponding vertex sets. We shall show that for each V either V ⊆ S or V ⊆ S c . Suppose V ∩ S = ∅. For each v in V ∩ S, there is a shortest directed path in D (A) from i to v whose weight is not divisible by p. Among all such v, choose w to have the shortest path π whose weight is not divisible by p. Consider x ∈ V . Since M is weakly connected, there is a sequence β of arcs in M such that underlying graph of β is a path from w to x. The choice of π implies that the arcs of π along with the arcs in β form a path α in PN (i, x) and wB (α) = wB (π) is not multiple of p. Thus x ∈ S, and we conclude that each vertex of V is in S. Now we show that each entry of B[S; S c ] is a multiple of p. Consider an arc (, m) with ∈ S and m ∈ S c . Then there exists γ ∈ PN (i, ) such that p wB (γ). Since m ∈ S c ,
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.7 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
7
γ does not contain the vertex m. Hence, the arcs of γ along with the arc (, m) form a path in PN (i, m) whose weight equals wB (γ)b,m . Since m ∈ S c this weight is a multiple of p. Since p is a prime which does not divide wB (γ), we conclude that p divides b,m . Hence each entry of B[S; S c ] is a multiple of p. By applying a permutation similarity if necessary, we may assume without loss of generality that S = {1, 2, . . . , |S|}. Let D be the diagonal matrix with di = 1/p for i = 1, . . . , |S|. Let C = DBD−1 . Since C is obtained from B by multiplying its first |S| rows (respectively, columns) by 1/p (respectively, p), C is an integer matrix and Ci,j = d/p. Thus C = DBD−1 is an integer matrix with (i, j)-entry d/p. Note that entries of C corresponding to N are all 1 because for each Vi , either Vi ⊆ S or Vi ⊆ S c . wC (γ) wB (γ) = for all β ∈ PN (i, j). Now by repeated By (2) and Lemma 2.4, d d/p application of the above technique can find a matrix which is diagonally similar to B whose entries corresponding to N are all 1, and whose (i, j) entry is ±1. These cases have been previously handled. Therefore, we conclude that B, and hence A, is INM . 2 3. Integrally normalizable patterns A zero–nonzero pattern A is a matrix A = [αi,j ] where αi,j ∈ {0, ∗}. A matrix A = [ai,j ] is an integer realization of A if ai,j = 0 when αi,j = 0 and ai,j is a non-zero integer when αi,j = ∗. The set of all integer realizations of A is denoted by QZ (A). Let A be an n × n zero–nonzero pattern and M be a subset of {(i, j) : i, j = 1, 2, . . . , n and i = j}. The zero–nonzero pattern A is integrally normalizable with respect to M if A is integrally normalizable with respect to M for each A ∈ QZ (A). Necessary and sufficient conditions for an n × n zero–nonzero pattern A with a spanning tree T in D (A) to be integrally normalizable with respect to the arcs of T are given in [2, Theorem 3.1]. In this section we give necessary and sufficient conditions for a zero–nonzero pattern A to be integrally normalizable with respect to a given arc set M of D (A), not just M corresponding to a spanning tree of D (A). The digraph D (A) = (V, Γ(A)) has vertex set V = {1, . . . , n} and arc set Γ(A) = {(i, j) : αi,j = ∗}. Since Γ(A) is a subset of {(i, j) : i, j = 1, 2, . . . , n}, we can view M as a subset of the non-loop arcs of D (A). We define CM (A) analogously, i.e., CM (A) = CM (A) for all A ∈ QZ (A). The following theorem gives a necessary condition for a zero–nonzero pattern to be integrally normalizable. Theorem 3.1. Let A be an n × n zero–nonzero pattern and M be a nonempty, non-loop arc set of Γ(A). If A is integrally normalizable with respect to M , then Γ(M ) is a forest.
JID:LAA 8
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.8 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
Proof. Assume A is integrally normalizable with respect to M . Then for each A ∈ QZ (A), there is B ∈ QZ (A) such that B = [bi,j ] is diagonally similar to A and bi,j = 1 for all (i, j) ∈ M . Suppose that Γ(M ) contains a cycle γ. Then by Lemma 2.4, for each A ∈ QZ (A), wB (γ F ) wA (γ F ) = = 1. R) R) wA (γM wB (γM Since this is not possible for all A ∈ QZ (A), Γ(M ) contains no cycles. Thus Γ(M ) is a forest. 2 The following theorem gives necessary and sufficient conditions for a zero–nonzero pattern to be integrally normalizable. Theorem 3.2. Let A be an n × n zero–nonzero pattern and M be a non-loop arc set of Γ(A). Then A is integrally normalizable with respect to M if and only if Γ(M ) is a forest and each γ in CM (A) is a directed cycle. Proof. First suppose that A is integrally normalizable with respect to M . By TheoR ) | wA (γ F ) rem 3.1, Γ(M ) is a forest. Let γ be a cycle in CM (A). By Theorem 2.5, wA (γM R = ∅, we can construct a matrix in A ∈ QZ (A) by assigning for each A ∈ QZ (A). If γM 1 to entries corresponding to γ F and integers greater than 1 to entries corresponding to R R R γM . Then wA (γM ) does not divide 1 = wA (γ F ). Therefore γM = ∅ and consequently γ is a directed cycle in CM (A). Conversely suppose that Γ(M ) is a forest and each γ in CM (A) is a directed cycle. Let A be in QZ (A). Since Γ(M ) is a forest, there is no cycle γ in CM (A) = CM (A) each of whose arcs lies in M . Thus (b) of Theorem 2.7 holds for A. Let γ be in CM (A) = CM (A). R R Then γ is a directed cycle and γM = ∅. Consequently 1 = wA (γM ) divides wA (γ F ). Thus by Theorem 2.7, A is INM . Since this is true for all A in QZ (A), A is integrally normalizable with respect to M . 2 4. Rational matrices whose cycles have integer weights In this section we present a result that is similar to a result of G. Engel and Hans Schneider regarding diagonal similarity. In addition the proof is similar to that of Theorem 2.7. In particular in [1], Engel and Schneider prove that the n × n complex matrix A is diagonally similar to a (0, 1)-matrix if and only if the weight of each cycle in D (A) is 1. Here we give necessary and sufficient conditions for a given rational matrix to be similar to an integer matrix. Theorem 4.1. Let A = [aij ] be a rational matrix of order n. Then A is diagonally similar to an integer matrix if and only if wA (γ) is an integer for each directed cycle in D (A).
JID:LAA
AID:13274 /FLA
[m1L; v1.159; Prn:15/07/2015; 15:49] P.9 (1-9)
S. Mallik, B.L. Shader / Linear Algebra and its Applications ••• (••••) •••–•••
9
Proof. First suppose that A is diagonally similar to an integer matrix B. Let γ be a directed cycle of D (A). The analog of Lemma 2.2 for rationals, implies wA (γ) = wB (γ), and hence wA (γ) is an integer. We prove the converse by induction on the number of entries of A that are non-integer. If each entry of A is an integer, then we are done. Assume that wA (γ) is an integer for each directed cycle in D (A), at least one entry of A is not an integer and the result holds for all such matrices with fewer non-integer entries. Let aij be a non-integer entry of A. Consider the matrix B obtained from A by replacing aij by 0. Then each directed cycle in D (B) is a directed cycle in D (A), and hence has an integer weight. Moreover, by induction B is diagonally similar to an integer matrix. Thus, A is diagonally similar to a matrix each of whose entries other than the (i, j)-entry is an integer. Without loss of generality, we may assume each entry of A other than the (i, j)-entry is an integer. Let r and s be relatively prime integers such that aij = r/s. Since |s| > 1, there exists a prime divisor p of s. Take S to be the set of all vertices v such that there exists a directed path from j to v in D (A) whose weight is not divisible by p. Then j ∈ S. Also, if there exists a directed path α from j to i, then α along with the arc (i, j) forms a directed cycle γ whose weight is an integer. As wA (γ) = wA (α)(r/s), we conclude that p / S. Now consider an arc of the form (u, v) where divides wA (α) for each such α. Thus, i ∈ u ∈ S and v ∈ / S. As u ∈ S, there exists a directed path β from j to u whose weight is an integer not divisible by p. The directed path β along with the arc (u, v) forms a directed path from j to v whose weight is an integer. As v ∈ / S the weight of this path must be divisible by the prime p. It follows that p divides auv . Moreover, each entry of A[S, S c ] is an integer divisible by p. Now consider the diagonal matrix D = diag(d1 , . . . , dn ) with dk = p if k ∈ S and dk = 1 otherwise. Then D−1 AD is matrix all of whose entries are integer except possibly the (i, j)-entry, and each of directed cycles in its digraph has integer weight. Note the (i, j) entry of D−1 AD is r/(s/p). If (s/p) = ±1, we are done. Otherwise, we replace A by D−1 AD and repeat the above process. As the denominators of the (i, j)-entries of the matrices obtained decrease in magnitude, we eventually end with an integer matrix that is diagonally similar to A. 2 References [1] G.M. Engel, H. Schneider, Cyclic and diagonal products on a matrix, Linear Algebra Appl. 7 (1973) 301–335. [2] C. Garnett, D.D. Olesky, B.L. Shader, P. van den Driessche, Integrally normalizable matrices and zero–nonzero patterns, Linear Algebra Appl. (2014) 132–153.