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Compounding Commuting Matrices Abstract 1 Introduction Journal of Advances in Mathematics and Computer Science 30(1): 1-8, 2019; Article no.JAMCS.45969 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851) Compounding Commuting Matrices ∗ Ronald P. Nordgren1 1Brown School of Engineering, Rice University, USA. Author's contribution The sole author designed, analyzed, interpreted and prepared the manuscript. Article Information DOI: 10.9734/JAMCS/2019/45969 Editor(s): (1) Dr. Zhenkun Huang, Professor, School of Science, Jimei University, China. Reviewers: (1) Abdullah Sonmezoglu, Bozok University, Turkey. (2) Grienggrai Rajchakit, Maejo University, Thailand. (3) Halil Grgn, Dicle University, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history/27948 Received: 27 September 2018 Accepted: 15 December 2018 Original Research Article Published: 24 December 2018 Abstract We formulate a method of constructing families of commuting matrices of increasing order by sequential compounding. Formulas are derived for sequentially constructing their Jordan forms and singular-value decompositions. Examples are given to illustrate our methods including construction of commuting Latin squares. Keywords: Commuting matrices; matrix compounding; Circulant Matrices; Latin squares; Jordan Form; SVD. 2010 Mathematics Subject Classification: 15B99; 00A06 1 Introduction Although matrices (as arrays) existed much earlier, matrix algebra originated with Arthur Cayley [1] in 1858 and was developed further by many others. As noted by Cayley, in general two matrices *Corresponding author: E-mail: [email protected]; Nordgren; JAMCS, 30(1): 1-8, 2019; Article no.JAMCS.45969 A and B do not commute, i.e.AB =6 BA. However, certain special families of matrices do commute, e.g. circulant matrices. Other families of commuting matrices are constructed in the present paper together with their Jordan form and singular-value decomposition (SVD). Our constructions employ the method of sequential compounding which has a long history as discussed by Rogers, et.al. [2] and used by them in constructions of magic squares and Latin squares. A great variety of commuting matrices can be formed by sequential compounding as our examples illustrate. The Jordan form and SVD of these commuting matrices are easy to construct, again by compounding. Our compounded commuting matrices should be regarded as complementary to circulant matrices for which formulas are available for their Jordan form. However, there are no known formulas for the SVD of circulant matrices and they do not compound to new circulant matrices as an example shows. Thus, our compounded matrices have certain advantages over circulant matrices. The Kronecker product, denoted by ⊗; and its associated identities (see Meyer [3]) are employed in our construction. All matrices in the present paper are square. 2 Compounding Constructions Definition 2.1. The matrix A is said to be commutal if its elements aij can be replaced by 0 0 0 new variables aij to form a matrix A of the same structural form as A such that A commutes with A: For example, circulant matrices are commutal, as this order-2 one illustrates: [ ] [ ] 0 0 c0 c1 0 c0 c1 0 0 C2 = ; C2 = 0 0 ; C2C2 = C2C2: (2.1) c1 c0 c1 c0 Note that C2 is a Latin square (c0 and c1 appear in all its rows and columns). Compounding. We compound members of two families of commutal matrices to form a third family of commutal matrices according to the following theorem: Let A 2 Rm×m be a commutal matrix and let B 2 Rn×n be a commutal matrix. Let 0 0 0 C = A ⊗ B; C = A ⊗B : (2.2) Then C 2 Rmn×mn is a commutal matrix. Proof. From (2.2) we have 0 ( 0 0) ( 0) ( 0) CC = (A ⊗ B) A ⊗B = AA ⊗ BB (2.3) ( 0 ) ( 0 ) ( 0 0) 0 = A A ⊗ B B = A ⊗B (A ⊗ B) = C C; whence CC0 = C0C, i.e. C is commutal. For example, compounding of two order-2 circulant matrices (2.1) gives the following commutal order-4 matrix: 2 3 [ ] [ ] a0b0 a0b1 a1b0 a1b1 6 7 a0 a1 b0 b1 6 a0b1 a0b0 a1b1 a1b0 7 C4 = ⊗ = 4 5 (2.4) a1 a0 b1 b0 a1b0 a1b1 a0b0 a0b1 a1b1 a1b0 a0b1 a0b0 2 Nordgren; JAMCS, 30(1): 1-8, 2019; Article no.JAMCS.45969 the elements of which can be relabeled as 2 3 c0 c1 c2 c4 6 7 6 c1 c0 c4 c2 7 C4 = 4 5 : (2.5) c2 c4 c0 c1 c4 c2 c1 c0 It can be verified directly that C4 is commutal. However, C4 is not a circulant matrix, i.e. two circulant matrices do not compound to a circulant matrix. It is noteworthy that C4 is a symmetric Latin square. Also, C4 is a representation of the Klein four-group (Vierergruppe) given by Felix Klein [4] in 1884. Jordan Form. The Jordan form of a matrix is of wide interest as discussed in many textbooks, e.g. Meyer [3]. It can be formed from the eigenvalues and eigenvectors of the matrix (when they m×m n×n exist). Here, the families of commutal matrices with members Ai 2 R and Bi 2 R are restricted to be diagonable with Jordan forms −1 −1 Ai = SADAiSA ; Bi = SB DBiSB ; (2.6) m×m n×n where DAi 2 C and DBi 2 C are diagonal matrices and the columns of the matrices SA m×m n×n 2 C and SB 2 C are their corresponding eigenvectors. Since Ai and Bi are diagonable and members of commuting families, it is known [3] that SA and SB are the same for all members of each family but DAi and DBi are different for each member of the family. The Jordan form of the compounded matrix C of (2.2) is given by the following theorem: The compounded commutal matrix Ci = Ai⊗Bi has the Jordan form −1 Ci = SC DCiSC ; (2.7) where ⊗ ⊗ −1 SC = SA SB ; DCi = DAi DBi = SC CiSC : (2.8) Proof. Using (2.6), we have ( ) ( ) −1 −1 Ci = Ai⊗B = SADAiS ⊗ SB DBiS i A ( ) B ⊗ ⊗ −1⊗ −1 = (SA SB )(DAi DBi) SA SB ; (2.9) −1 ⊗ −1 −1⊗ −1 SC = (SA SB ) = SA SB : −1 DCi is diagonal since DAi and DBi are. The relation DCi = SC CiSC follows from (2.7). ( ) T T T T Furthermore, if Ai and Bi are normal matrices AiAi = Ai Ai; BiBi = Bi Bi ; it can be shown that Ci is normal and [3] −1 ∗ −1 ∗ −1 ∗ SA = SA; SB = SB ; SC = SC ; (2.10) ∗ where S denotes the conjugate transpose of S: In addition, if Ai and Bi are symmetric matrices (always normal), then [3] −1 T −1 T −1 T SA = SA; SB = SB ; SC = SC ; (2.11) i.e. they are orthogonal and Ci is symmetric too. When Ai and Bi are real and symmetric, all S's and D's are real too. When Ai and Bi also are positive definite or positive semi-definite , so is Ci: For example, in the Jordan form for A2 of (2.1), [ ] 1 1 1 SA = SB = p ; DA = diag (a0 ± a1) ; DB = diag (b0 ± b1) ; (2.12) 2 1 −1 3 Nordgren; JAMCS, 30(1): 1-8, 2019; Article no.JAMCS.45969 and in the Jordan form for C4 of (2.5), by (2.8), 2 3 [ ] [ ] 1 1 1 1 6 − − 7 ⊗ p1 1 1 ⊗ p1 1 1 1 6 1 1 1 1 7 SC = SA SB = = 4 5 ; 2 1 −1 2 1 −1 2 1 1 −1 −1 1 −1 −1 1 ⊗ −1 DC = DA DB = SC C4SC = diag [λ1; λ2; λ3; λ4] ; (2.13) where λ1 = c0 + c1 + c2 + c4; λ2 = c0 − c1 + c2 − c4; λ3 = c0 + c1 − c2 − c4; λ4 = c0 − c1 − c2 + c4: The Jordan form of C4 can be verified directly from (2.13). Other examples of the Jordan form construction are given in what follows. SVD. The singular-value decomposition (SVD) of a matrix also is of general interest. It has many applications as discussed by Martin and Porter [5] and Meyer [3], where additional references are given. We first review the known facts concerning the SVD and then we consider the SVD for compounded commutal matrices. The SVD of any matrix A 2 Rm×m reads A = UΣVT ; (2.14) ( ) where U 2 Rm×m and V 2 Rm×m , are orthogonal matrices UUT = VVT = I and Σ 2 Rm×m is a diagonal matrix with r = rank (A) positive diagonal elements σk (called the singular values) and m − r zero elements. From (2.14) we have AAT = UΣ2UT ; AT A = VΣ2VT (2.15) from which U and V can be determined subject to (2.14) being satisfied. For example, if U and Σ2 are obtained form (2.15) and A is nonsingular (r = m), then V is given by − V = AT UΣ 1 (2.16) which can be shown to be orthogonal. If A is singular, then other considerations are necessary, see Meyer [3]. The construction of V will not be mentioned in the examples that follow. If A is normal, then by (2.6) and (2.10) the Jordan form of A reads ∗ ∗ ∗ A = SDS and AAT = AA = S jDj2 S ; (2.17) where the (possibly complex) eigenvalues of D are denoted by λk: From (2.15) and (2.17) we see that the unique eigenvalues of AAT are contained in both Σ2 and jDj2 ; thus σk = jλkj : (2.18) However, when S is complex it cannot be used for U (nor S∗ for VT ). For symmetric (normal) matrices, S and D are real and S is orthogonal, hence U = S is allowed. If A is symmetric and positive semi-definite , then λk ≥ 0 and the Jacobi form and SVD of A are identical.
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