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 ≠ 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 ′ ′ ′ 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: [ ] [ ] ′ ′ c0 c1 ′ c0 c1 ′ ′ C2 = , C2 = ′ ′ , 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 ∈ Rm×m be a commutal matrix and let B ∈ Rn×n be a commutal matrix. Let

′ ′ ′ C = A ⊗ B, C = A ⊗B . (2.2)

Then C ∈ Rmn×mn is a commutal matrix.

Proof. From (2.2) we have

′ ( ′ ′) ( ′) ( ′) CC = (A ⊗ B) A ⊗B = AA ⊗ BB (2.3) ( ′ ) ( ′ ) ( ′ ′) ′ = A A ⊗ B B = A ⊗B (A ⊗ B) = C C, whence CC′ = C′C, i.e. C is commutal.

For example, compounding of two order-2 circulant matrices (2.1) gives the following commutal order-4 matrix:   [ ] [ ] a0b0 a0b1 a1b0 a1b1   a0 a1 b0 b1  a0b1 a0b0 a1b1 a1b0  C4 = ⊗ =   (2.4) a1 a0 b1 b0 a1b0 a1b1 a0b0 a0b1 a1b1 a1b0 a0b1 a0b0

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the elements of which can be relabeled as   c0 c1 c2 c4    c1 c0 c4 c2  C4 =   . (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 ∈ R and Bi ∈ R are restricted to be diagonable with Jordan forms

−1 −1 Ai = SADAiSA , Bi = SB DBiSB , (2.6)

m×m n×n where DAi ∈ C and DBi ∈ C are diagonal matrices and the columns of the matrices SA m×m n×n ∈ C and SB ∈ 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 = √ , DA = diag (a0 ± a1) , DB = diag (b0 ± b1) , (2.12) 2 1 −1

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and in the Jordan form for C4 of (2.5), by (2.8),   [ ] [ ] 1 1 1 1  − −  ⊗ √1 1 1 ⊗ √1 1 1 1  1 1 1 1  SC = SA SB = =   , 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 ∈ Rm×m reads

A = UΣVT , (2.14) ( ) where U ∈ Rm×m and V ∈ Rm×m , are orthogonal matrices UUT = VVT = I and Σ ∈ 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 |D|2 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 |D|2 , thus

σk = |λk| . (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. ( ) m×m T T Now, let Ai ∈ R be a member of a family of normal commutal matrices AiAi = Ai Ai . T T T T It follows that AiAj = Aj Ai for all i, j and it can be shown that AiAi and Aj Aj commute. Therefore, UAi in the SVD of Ai is the same for all Ai (UAi = UA), hence

T 2 T AiAi = UAΣAiUA (2.19)

The SVD also applies to rectangular and complex martices, see [5, 3].

4 Nordgren; JAMCS, 30(1): 1-8, 2019; Article no.JAMCS.45969

and we have the following theorem: For Ai and Bi members of families of normal commutal matrices, the compounded commutal matrix Ci = Ai⊗Bi has the SVD T Ci = UC ΣCiVCi, (2.20) where

( ) 1 T T 2 T T T UC = UA⊗UB , ΣCi = ΣAi⊗ΣBi = UAAiAi UA , VCi = VAi⊗VBi. (2.21)

Proof. ( ) ( ) T T Ci = Ai⊗Bi = UAΣAiVAi ⊗ UB ΣBiVBi ( ) T T T = (UA⊗UB )(ΣAi⊗ΣBi) VAi⊗VBi = UC ΣCiVCi. (2.22)

ΣCi is diagonal since ΣAi and ΣBi are. It can be shown that UC and VCi are orthogonal. The ( ) 1 T T 2 relation ΣCi = UAAiAi UA follows from (2.19).

As already noted, Ci is normal when Ai and Bi are normal and thus its singular values are related to its eigenvalues by a relation of the form (2.18). From (2.21) and (2.8) we have

∗ ∗ ∗ ∗ UC = (UASASA) ⊗ (UB SB SB ) = (UASA) ⊗ (UB SB )(SA⊗SB ) , (2.23) i.e. the SVD matrix UC is related to the eigenvector matrix SC by

∗ ∗ ∗ UC = LSC , L = (UASA) ⊗ (UB SB ) LL = I. (2.24)

Examples of the SVD construction are given in what follows.

3 Examples

Example 1. Here is an order-8 normal commutal matrix constructed from compounding C4 of (2.5) with C2 of (2.1) and relabeling the elements as before:   c0 c1 c2 c3 c4 c5 c6 c7    c1 c0 c3 c2 c5 c4 c7 c6     c2 c3 c0 c1 c6 c7 c4 c5     c3 c2 c1 c0 c7 c6 c5 c4  C8 =   . (3.1)  c4 c5 c6 c7 c0 c1 c2 c3     c5 c4 c7 c6 c1 c0 c3 c2    c6 c7 c4 c5 c2 c3 c0 c1 c7 c6 c5 c4 c3 c2 c1 c0

Again, it is noteworthy that C8 is a symmetric Latin square. In view of its symmetry, UC8 =

SC8 and σi = |λi|. Furthermore, repeated application of our compounding construction using (2.1) produces symmetric Latin squares of order 2k. Also, commutal normal Latin squares of any nonprime order can be produced by compounding two circulant matrices which are commutal and normal. The Jordan form and SVD of such matrices can be constructed sequentially from our formulas.

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Example 2. We construct an order-6 commutal matrix by compounding an order-3 commutal matrix with an order-2 commutal matrix. We start with the symmetric order-3 commutal matrix   a0 a1 a2   A = a1 a0 + a2 a1 . (3.2) a2 a1 a0 The Jordan form of A has  √  − 2 1 1 ( ) 1 √ √ √ S =  0 2 − 2  , D = diag a − a , a + a ± 2a . (3.3) A 2 √ A 0 2 0 2 1 2 1 1

From DA we see that A with integer elements is nonsingular (λi ≠ 0) unless√a0 = a2 or a0 = −a2, a1 = 0. Furthermore, A is positive semi-deﬁnite if a0 ≥ a2 and a0 + a2 ≥ ± 2a1. The SVD of A has UA = SA, ΣA = diag (σ1, σ2, σ3) , σi = |λi| . (3.4) Next, we introduce the following order-2 commutal normal matrix: [ ] b b B = 0 1 (3.5) −b1 b0 whose Jordan form has √ [ ] √ [ ] 2 1 1 ∗ 2 1 i S = , [S ] = , D = diag (b ∓ ib ) . (3.6) B 2 −i i B 2 1 −i B 0 1 From (2.19), the SVD of B has

UB = VB = I2, ΣB = diag (σ1, σ2) , ( ) 1 2 2 2 σ1 = σ2 = b0 + b1 = |λ1| = |λ2| . (3.7) On compounding A⊗B we obtain the order-6 commutal matrix C which after relabeling its elements reads   c0 c1 c2 c3 c4 c5    −c1 c0 −c3 c2 −c5 c4     c2 c3 c0 + c4 c1 + c5 c2 c3  C =   . (3.8)  −c3 c2 −c1 − c5 c0 + c4 −c3 c2    c4 c5 c2 c3 c0 c1 −c5 c4 −c3 c2 −c1 c0 By (2.8), the Jordan form of C has  √ √ √ √  −2 −2 √2 √2 √2 √2    2i −2i −i 2 i 2 −i 2 i 2    1  0 0 2 2 −2 −2  −1 ∗ SC =   , SC = SC , 4  0 0 −√2i √2i √2i −√2i    2 2 √2 √2 √2 √2 −2i 2i −i 2 i 2 −i 2 i 2

DC = diag (λ1, λ2, . . . , λ6) , (3.9) where

λ1 = c0 − c4 − i (c1 − c5) , λ2 = λ¯1, √ ( √ ) λ3 = c0 + c4 + 2c2 − i c1 + c5 + 2c3 , λ4 = λ¯3, √ ( √ ) λ5 = c0 + c4 − 2c2 − i c1 + c5 − 2c3 , λ6 = λ¯5,

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as can be veriﬁed directly. From DC , we see that C with integer elements is nonsingular unless c0 = c4, c1 = c5 or c0 = −c4, c1 = −c5, c2 = c3 = 0. By (2.21) with (3.4) and (2.20), the SVD of C has  √  − 2√ 0 1 0 1 0    0 − 2√ 0 1√ 0 1    1  0 0 2√ 0 − 2√ 0  UC = S ⊗ I2 =   . (3.10) 2  0 0 0 2 0 − 2   √  2√ 0 1 0 1 0 0 2 0 1 0 1 By (2.19) with (3.10), the singular values of C are

[ ] 1 T T 2 ΣC = UC AC AC UC = diag (σ1, σ1, σ3, σ3, σ5, σ5,) , σk = |λk| , k = 1, 3, 5, (3.11) as can be veriﬁed directly. The relation between σk and λk follows from (2.18) since C is a normal matrix (because A and B are normal). However, SC is complex and therefore it cannot be used as

UC instead of (3.10). The relation (2.24) between SC and UC has   1 i 0 0 0 0   √  1 −i 0 0 0 0    ∗ ∗ ∗ 2  0 0 1 i 0 0  LC = (UASA) ⊗ (UB SB ) = I3 ⊗ SB =   . (3.12) 2  0 0 1 −i 0 0   0 0 0 0 1 i  0 0 0 0 1 −i There are many other possibilities for constructing an order-6 commutal matrix using various combinations of (3.2), (2.1), (2.5), and an order-3 circulant matrix. One could start with an order-2 matrix and use two order-3 commutal matrices in the construction. There are endless possibilities for constructing higher-order commutal matrices sequentially by repeated compounding as the reader should enjoy doing. Example 3. For comparison purposes we record the Jordan form of the order-4 circulant matrix   c0 c3 c2 c1   ˆ  c1 c0 c3 c2  C4 =   . (3.13) c2 c1 c0 c3 c3 c2 c1 c0 From known formulas [3], we ﬁnd that ˆ ˆ ˆ ˆ∗ C4 = S4D4S4, (3.14) where   1 1 1 1 1  1 i −1 −i  Sˆ =   , Dˆ = diag (λ , λ , λ , λ ) , 4 2  1 −1 1 −1  4 1 2 3 4 1 −i −1 i

λ1 = c0 + c1 + c2 + c3, λ2 = c0 − c2 − i (c1 − c3) , (3.15)

λ3 = c0 − c1 + c2 − c3, λ4 = c0 − c2 + i (c1 − c3) .

In addition, the SVD of Cˆ 4 is found to be ˆ ˆ ˆ ˆ T C4 = U4Σ4V4 , (3.16)

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where, on noting that |λ2| = |λ4| and recombining their associated eigenvectors in Sˆ4, we ﬁnd that  √  1 2 1√ 0  − −  ˆ 1  1√ 0 1 2  ˆ ˆ T U4 =   , U4U4 = I, (3.17) 2 1 − 2 1√ 0 1 0 −1 2

Σˆ 4 = diag (σ1, σ2, σ3, σ4) , σk = |λk| .

However, formulas for the SVD of a circulant matrix of arbitrary order are lacking. Perhaps further research will discover them.

4 Conclusions

A large variety of families of commuting matrices, their Jordan form, and SVD can be constructed by the compounding methods presented herein. Our examples illustrate the possibilities, including families of commuting Latin squares. Competing Interests

Author has declared that no competing interests exist.

References

[1] Cayley A. A memoir on the theory of matrices. Phil. Trans. 1858;148:17-37. Available:https://archive.org/stream/philtrans05474612/05474612#page/n0/mode/2up.

[2] Rogers A, Cameron I, Loly P. Compounding doubly aﬃne matrices; 2017. Available:https://arxiv.org/abs/1711.11084

[3] Meyer CD. Matrix analysis and applied linear algebra. Society for Industrial and Applied Mathematics; 2000.

[4] Klein F. Lectures on the Icosahedron and the solution of equations of the ﬁfth degree. (1884), Translated by GG Morrice, 2nd and rev. ed. Kegan Paul, Trench, Tr¨ubner& Co. Ltd.; 1913.

[5] Martin C, Porter M. The extraordinary SVD. MAA Monthly. 2012;119:838-851.

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