Linear and Multilinear Algebra ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/glma20 Block determinants, partial determinants and the exponential map A. Fošner, Y. Hardy & B. Zinsou To cite this article: A. Fošner, Y. Hardy & B. Zinsou (2024) Block determinants, partial determinants and the exponential map, Linear and Multilinear Algebra, 72:17, 2879-2914, DOI: 10.1080/03081087.2024.2303069 To link to this article: https://doi.org/10.1080/03081087.2024.2303069 © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. Published online: 18 Jan 2024. 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Fošnera, Y. Hardy b,c and B. Zinsoub,c aFaculty of Management, University of Primorska, Koper, Slovenia; bSchool of Mathematics, University of the Witwatersrand, Johannesburg, South Africa; cNational Institute for Theoretical and Computational Sciences (NITheCS), Johannesburg, South Africa ABSTRACT The block trace and partial trace are closely related. However, the block determinant and partial determinant are quite different. We introduce a new determinant-like operation, the determinant-root, which has a partial operation that is more natural in the algebraic sense. The block determinant retains some of the algebraic prop- erties of determinants, while the partial determinant-root retains the same algebraic properties for matrices which are Kronecker products.We investigate the equationsdet(D(A)) = det(A),D(AB) = D(A)D(B) and D(eA) = eT(A), where D is the block/partial/partial- root determinant and T is the corresponding block/partial/partial- normalized trace. These results yield a characterization of non-linear preservers of determinants of Kronecker products. ARTICLE HISTORY Received 30 November 2022 Accepted 29 August 2023 COMMUNICATED BY E. Poon KEYWORDS Kronecker product; Kronecker sum; block determinant; exponential map; partial determinant; partial trace MATHEMATICS SUBJECT CLASSIFICATIONS 15A15; 15A30; 15A69 1. Introduction Let {E[n]ij : i, j ∈ {1, . . . , n} } denote the standard basis in Mn(F), the algebra of n × n matrices over a field F. In other words, E[n]ij is an n × n matrix with a 1 in the i-th row and the j-th column and all other entries are zero. The standard basis is given entry-wise by (E[n]ij )kl = δikδjl, where δik is the Kronecker delta δik = { 1; i = k, 0; i �= k. The transpose of thematrixA ∈ Mn(F)will be denoted byAT . Let⊗ denote the Kronecker product of matrices, i.e. for A ∈ Mm(F) and B ∈ Mn(F) the matrix A ⊗ B is given block- wise by A ⊗ B := ⎛ ⎜⎜⎜⎝ (A)11B (A)12B · · · (A)1mB (A)21B (A)22B · · · (A)2mB ... ... . . . ... (A)m1B (A)m2B · · · (A)mmB ⎞ ⎟⎟⎟⎠ , CONTACT A. Fošner ajda.fosner@fm-kp.si Faculty of Management, University of Primorska, Izolska vrata 2, SI-6000 Koper, Slovenia © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. http://www.tandfonline.com https://crossmark.crossref.org/dialog/?doi=10.1080/03081087.2024.2303069&domain=pdf&date_stamp=2024-11-07 http://orcid.org/0000-0003-0264-8874 mailto:ajda.fosner@fm-kp.si http://creativecommons.org/licenses/by/4.0/ 2880 A. FOŠNER ET AL. where (A)ij is the entry of A in the i-th row and j-th column. The Kronecker sum of A ∈ Mm(F) and B ∈ Mn(F) is the matrix A ⊕ B = A ⊗ In + Im ⊗ B, where Ik is the identity matrix inMk(F), k = m, n. We have Mmn(F) = Mm(Mn(F)) = Mm(F)⊗ Mn(F). Let A ∈ Mmn(F), then we may write A = n∑ i,j=1 Aij ⊗ E[n]ij , A = m∑ i,j=1 E[m] ij ⊗ A′ ij, where A′ ij ∈ Mn(F) are the blocks of A, and Aij ∈ Mm(F) are the ‘block like’ parts of A induced by the isomorphismMm(F)⊗ Mn(F) = Mn(F)⊗ Mm(F). Choi gives the definition of the partial determinants det1(A) and det2(B) as follows. Definition 1.1 ([1]): Let A ∈ Mmn(F) with A = n∑ i,j=1 Aij ⊗ E[n]ij , A = m∑ i,j=1 E[m] ij ⊗ A′ ij. The partial determinants det1(A) and det2(A) are given by det1(A) := n∑ i,j=1 det(Aij)E [n] ij , det2(A) := m∑ i,j=1 det(A′ ij)E [m] ij . This definition is analogous to that of the partial trace. The partial trace features prominently in quantum information theory (see for example [2]). Definition 1.2: Let A ∈ Mmn(F) with A = n∑ i,j=1 Aij ⊗ E[n]ij , A = m∑ i,j=1 E[m] ij ⊗ A′ ij. The partial traces tr1(A) and tr2(A) are given by tr1(A) := n∑ i,j=1 tr(Aij)E [n] ij , tr2(A) := m∑ i,j=1 tr(A′ ij)E [m] ij . Proposition 1.1: Let r ∈ N, A1, . . . ,Ar ∈ Mm(F) and B1, . . . ,Br ∈ Mn(F). Then tr1 ⎛ ⎝ r∑ j=1 Aj ⊗ Bj ⎞ ⎠ = r∑ j=1 tr(Aj)Bj, and tr2 ⎛ ⎝ r∑ j=1 Aj ⊗ Bj ⎞ ⎠ = r∑ j=1 tr(Bj)Aj. LINEAR ANDMULTILINEAR ALGEBRA 2881 Proof: We have tr1 ⎛ ⎝ r∑ j=1 Aj ⊗ Bj ⎞ ⎠ = tr1 ⎛ ⎝ r∑ j=1 n∑ k,l=1 Aj ⊗ ((Bj)klE [n] kl ) ⎞ ⎠ = tr1 ⎛ ⎝ r∑ j=1 n∑ k,l=1 ((Bj)klAj)⊗ E[n]kl ⎞ ⎠ = r∑ j=1 n∑ k,l=1 tr((Bj)klAj)E [n] kl = r∑ j=1 tr(Aj) n∑ k,l=1 ((Bj)klE [n] kl ) = r∑ j=1 tr(Aj)Bj. � The following proposition shows that tr1 is the block trace of [3], i.e. the trace ofA taken over the matrix ringMn(F). Proposition 1.2: Let A ∈ Mmn(F) with A = m∑ i,j=1 E[m] ij ⊗ A′ ij. Then tr1(A) = A′ 11 + A′ 22 + · · · + A′ mm. Proof: We have, by Proposition 1.1, tr1(A) = m∑ i,j=1 tr(E[m] ij )A′ ij = m∑ i=1 A′ ii. � Consequently, we note the following corollary for any A ∈ Mmn(F) and B ∈ Mn(F). Corollary 1.1: Let A ∈ Mmn(F), B ∈ Mn(F) and C ∈ Mm(F). Then tr1(A(Im ⊗ B)) = tr1(A)B and tr2(A(C ⊗ In)) = tr2(A)C. Proof: We have tr1(A(Im ⊗ B)) = tr1 ⎛ ⎝ ⎛ ⎝ m∑ i,j=1 E[m] ij ⊗ A′ ij ⎞ ⎠ (Im ⊗ B) ⎞ ⎠ 2882 A. FOŠNER ET AL. = tr1 ⎛ ⎝ m∑ i,j=1 (E[m] ij Im)⊗ (A′ ijB) ⎞ ⎠ = tr1 ⎛ ⎝ m∑ i,j=1 E[m] ij ⊗ (A′ ijB) ⎞ ⎠ = m∑ i,j=1 tr(E[m] ij )(A′ ijB) = m∑ j=1 A′ jjB = tr1(A)B, and similarly, tr2(A(C ⊗ In)) = tr2(A)C. � The partial trace is linear while the partial determinant is not linear. Additionally, the partial trace is ‘partial’ in the sense that the trace can be completed, i.e. using Proposi- tion 1.1, tr(tr1(A ⊗ B)) = tr(tr(A)B) = tr(A) tr(B) = tr(A ⊗ B), so that, by linearity of the trace and partial trace, tr(tr1(A)) = n∑ i,j=1 tr(tr1(Aij ⊗ E[n]ij )) = n∑ i,j=1 tr(Aij ⊗ E[n]ij ) = tr(A). Similarly, tr(tr1(A)) = tr(tr2(A)) = tr(A). Thompson showed that if A ∈ Mmn(C) is positive definite, then det(det2(A)) ≥ det(A) and that det(det2(A)) = det(A) ⇐⇒ A′ ij = δijA′ ij, where δij denotes the Kronecker delta [4]. Let R := Mn(F) be the associative ring of n × n matrices over F. Then the trace trR in Mm(R) is the block trace, i.e. trR(A) = tr1(A). The same relation does not hold for the block determinant detR, if we use the definition of detR from [5, Definition 2.1]. Definition 1.3: Let B ∈ Mm(R), where R = Mn(F), with B = m∑ i,j=1 E[m] ij ⊗ Bij. The block determinant detR over R is given by detR(B) = ∑ σ∈πm sgn(σ )B1σ(1) · · ·Bmσ(m), LINEAR ANDMULTILINEAR ALGEBRA 2883 where the sum is taken over all permutations σ of { 1, 2, . . . ,m }, i.e. the symmetric group πm. Here, det(detR(B)) = det(B) when the blocks of B commute [6]. In fact, not all of the blocks need commute as shown in [5, Theorem 1.2]. Another important property to consider is whether the identity det(AB) = det(A) det(B) carries over to the partial determinant. In other words, what are the conditions onA andB such that det1(AB) = det1(A) det1(B)? The same question can be asked for the block determinant. In the case of the block determinant, we have detR(AB) = detR(A) detR(B) (Theorem 2.1) under certain commutativity conditions on the entries of A and B in R. SinceB ⊗ C is a permutation similar toC ⊗ B via the vec-permutationmatrix P (perfect shuffle matrix) [7, 8] so that det2(A) = det1(PAPT), we confine our attention to det1(A). The characterization for arbitrary matrices (i.e. neither Kronecker products nor posi- tive definite matrices) remains open. We also consider a (less restricted) monoid Dm(F) and a determinant-root operation Det (see Section 4.2 of this text), such that the partial determinant-root obeys Det(A ⊗ B) = Det(Det1(A ⊗ B)) and Det1((A ⊗ B)(C ⊗ D)) = Det1(A ⊗ B)Det1(C ⊗ D). Finally, we note the connection between the determinant and trace. If F is the field of complex numbers or the field of real numbers and A ∈ Mn(F), then det(eA) = etr(A). (1) The exponential map also provides a connection between Kronecker sums and Kronecker products [9, p. 440], i.e. eA⊕B = eA ⊗ eB. We investigate whether the exponential-determinant-trace relation may hold for det1 and tr1. In the case of Kronecker sums, we have a straightforward answer. We also show that detR(eA) = etrR(A) in a certain factor group, and that for a Kronecker sum A, we have Det1(eA) = eTr1(A) over a certain factor group. Here, Tr(A) = 1/n tr(A) in Mn(F) and Tr1(A) = 1/m tr1(A) inMmn(F). For partial determinants and partial determinant-roots of arbitrary matrices (see Section 4.2 of this text), we have not found an equivalent result. Finally, we note that det1(Imn) = In, detR(Imn) = In and Det1(Imn) = In. WhenF is the field of real numbers or the field of complex numbers, defineHn ⊂ Mn(F) to be the Hermitian matrices in Mn(F). Linear maps preserving properties of Kronecker products of matrices have received considerable attention in recent years. Such maps are closely connected to quantum information science (see, e.g. [10]).More recently, Ding et al. considered linear preservers of determinants of Kronecker products of Hermitianmatrices [11], i.e. linear maps φ : Hmn → Hmn satisfying det(φ(A ⊗ B)) = det(A ⊗ B), where A and B are Hermitian. A few of the results in [11] are restricted to the case when A and B are positive or negative semidefinite matrices. In order to study this problem more 2884 A. FOŠNER ET AL. generally, we make use of the identity (1). In particular, det ( eA ⊗ eB ) = etr(A⊕B), where ⊕ is the Kronecker sum, i.e. A ⊕ B := A ⊗ In + Im ⊗ B, where Ik, k = m, n denotes the k × k identity matrix. Under exponentiation of Hermitian matrices, the Kronecker sum arises naturally as the unique map ⊕ : Hm × Hn → Hm ⊗ Hn satisfying eA⊕B = eA ⊗ eB, A ∈ Hm, B ∈ Hn. Moreover, A ∈ Mn(C) is non-singular if and only if A = eB for some B ∈ Mn(C) [9, Example 6.2.15]. Thus, in studying linear mapsψ : Mmn(F) → Mmn(F) preserving deter- minants of (non-singular) Kronecker products det(ψ(A ⊗ B)) = det(A ⊗ B) of non-singular matrices A and B, it suffices to study maps φ : Mmn → Mmn preserving the trace of Kronecker sums tr(φ(A ⊕ B)) = tr(A ⊕ B). (2) In this paper we will confine our attention to linear maps φ satisfying (2). We denote by GLn(F) and SLn(F) the general and special linear groups inMn(F) respectively. Thus, we are also interested in non-linear preservers of the determinant of Kronecker product in GLmn(F) in terms of linear preservers of the Kronecker sum. Non-linear preservers of the determinant have also been studied in [12], where preservers were observed under the additive structure ofMn(F). We also investigate preservers in terms of additive structure, but due to the indirect (via exponentiation) approach, we are able to reframe the study in terms of linear preservers. 2. Block determinants Note that det(detR(B)) = det(B) when the blocks of B commute [6] or, more generally, under a sufficient commutativity condition [5]. Now we turn our attention to products and to the exponential map. 2.1. Products Bourbaki [13, §6 Theorem 1] proved Theorem 2.1 in terms of exterior products. For completeness, we provide a proof using Definition 1.3. Theorem 2.1: Let A,B ∈ Mmn(F) and let R be a commutative subring of Mn(F). Then detR(AB) = detR(A) detR(B). LINEAR ANDMULTILINEAR ALGEBRA 2885 Proof: Since all of the blocks of A and B in R commute, the proof proceeds exactly as in the case of matrices over a field. We have detR(A) detR(B) = ∑ σ ,ψ∈πm sgn(σ ) sgn(ψ)A1,σ(1) · · ·Am,σ(m)B1,ψ(1) · · ·Bm,ψ(m) = ∑ σ ,ψ∈πm sgn(σ ) sgn(ψ)A1,σ(1) · · ·Am,σ(m)Bσ(1),ψ◦σ(1) · · ·Bσ(m),ψ◦σ(m) = ∑ σ ,ψ ′∈πm sgn(σ ) sgn(ψ ′ ◦ σ−1)A1,σ(1) · · ·Am,σ(m)Bσ(1),ψ ′(1) · · ·Bσ(m),ψ ′(m) = ∑ σ ,ψ ′∈πm sgn(σ ) sgn(ψ ′ ◦ σ−1)A1,σ(1)Bσ(1),ψ ′(1) · · ·Am,σ(m)Bσ(m),ψ ′(m) = ∑ σ ,ψ ′∈πm sgn(ψ ′)A1,σ(1)Bσ(1),ψ ′(1) · · ·Am,σ(m)Bσ(m),ψ ′(m). On the other hand, detR(AB) = ∑ ψ ′∈πm sgn(ψ ′) (AB)1,ψ ′(1) · · · (AB)1,ψ ′(m) = ∑ ψ ′∈πm m∑ k1,...,km=1 sgn(ψ ′)A1,k1Bk1,ψ ′(1) · · ·Am,kmBkm,ψ ′(m) and the terms in this sum are zero whenever ki = kj, i �= j. Let us consider the terms where k1 = k2. Then, since every permutation can be expressed as an even permutation composed with either the identity or the permutation which exchanges 1 and 2, we have detR(AB) = ∑ ψ ′∈πm m∑ k1,...,km=1 sgn(ψ ′)A1,k1Bk1,ψ ′(1) · · ·Am,kmBkm,ψ ′(m) = ∑ ψ ′∈πm sgn(ψ ′)=1 m∑ k1,...,km=1 sgn(ψ ′) [ A1,k1Bk1,ψ ′(1)A2,k2Bk2,ψ ′(2) · · ·Am,kmBkm,ψ ′(m) − A1,k1Bk1,ψ ′(2)A2,k2Bk2,ψ ′(1) · · ·Am,kmBkm,ψ ′(m) ] = ∑ ψ ′∈πm sgn(ψ ′)=1 sgn(ψ ′) m∑ k1,...,km=1 k1 �=k2 A1,k1Bk1,ψ ′(1)A2,k2Bk2,ψ ′(2) · · ·Am,kmBkm,ψ ′(m) = ∑ ψ ′∈πm sgn(ψ ′) m∑ k1,...,km=1 k1 �=k2 A1,k1Bk1,ψ ′(1) · · ·Am,kmBkm,ψ ′(m). 2886 A. FOŠNER ET AL. Proceeding in the same way for all i �= j, the sum is taken over all terms with distinct k1, k2, . . . , kn, i.e. the sum is over all permutations σ , where we identify σ(j) = kj, detR(AB) = ∑ ψ ′,σ∈πm sgn(ψ ′)A1,σ(1)Bσ(1),ψ ′(1) · · ·Am,σ(m)Bσ(m),ψ ′(m). � 2.2. The exponential map Note that exp F (A) = expR(A), where R = Mn(F), i.e. the power series for eA yields the same matrix with multiplication in the field F and multiplication in the ring R. Our first proposition shows that, in the special case of Kronecker sums A, the block trace and block determinant obey detR(eA) = etrR(A). Proposition 2.1: Let F be the field of real numbers or the field of complex numbers. Let A ∈ Mm(F), B ∈ Mn(F) and let R be the matrix ring R = Mn(F). Then detR(eA⊕B) = etrR(A⊕B). Proof: Since all of the blocks (eA)ijeB of eA ⊗ eB commute, detR(eA⊕B) = detR(eA ⊗ eB) = (eB)m detF(eA) = emBetr(A) and etrR(A⊕B) = etr(A)In+mB = emBetr(A). � In the following, we let R be a commutative ring, and consider the matrix ringMm(R). In particular, we will consider the case R ⊂ Mn(F), i.e. R is a commutative subring of the n × nmatrices over F, and hence we will be considering block structured matrices in Mm(R) ⊂ Mm(Mn(F)). We have a weaker result of Proposition 2.1 for matrices which are not Kronecker sums. Let R be a commutative ring and let [Mm(R),Mm(R)] denote the R- submodule ofMm(R) generated by all commutators inMm(R). If trR(A) = 0n, then A is in an R-submodule of [Mm(R),Mm(R)] [14, Theorem 15]. The next lemma shows that matri- ces with zero trace, have unit determinant under the exponential map. By Theorem 2.1 we have detR(eA) detR(eB) = detR(eAeB) Now eAeB is non-singular, so there exists C ∈ Mm(R) such that eAeB = eC [9, Example 6.2.15]. Thus, if R is a commutative subring ofMn(F), then the set G = { detR(eA) : A ∈ Mm(R) } is an abelian multiplicative group in R with In = detR(e0mn) and (detR(eA))−1 = detR(e−A). LINEAR ANDMULTILINEAR ALGEBRA 2887 Lemma 2.1: Let F be the field of real numbers or the field of complex numbers. Let A ∈ ker(trR), where R is a commutative subring (with identity) of Mn(F). Then KA detR(eA) = KA, (3) where KA is the normal subgroup of G which is multiplicatively generated by A, i.e. KA = { detR(ejA) : j ∈ Z }. Furthermore, KA is the smallest subgroup of G which satisfies Equation (3). Proof: Clearly In ∈ KA and (detR(eA))−1 = detR(e−A) ∈ KA. By construction, KA is closed. Since R is commutative, KA is a multiplicative normal subgroup of G. If K ′ ⊆ G satisfies Equation (3), then KA ⊆ K ′ since In ∈ K ′. � Theorem 2.2: Let F be the field of real numbers or the field of complex numbers. Let A ∈ Mm(R), where R is a commutative subring (with identity) of Mn(F) such that exp( 1m trR(A)) ∈ R. Then KTZ(A) detR(eA) = KTZ(A)etrR(A), where TZ(A) := A − 1 m Im ⊗ trR(A), and KTZ(A) := { detR(ejTZ(A)) : j ∈ Z }. Proof: Since R is commutative, TZ(A) and 1 mIm ⊗ trR(A) commute inMm(R) and eTZ(A)e 1 m Im⊗trR(A) = eTZ(A)+ 1 m Im⊗trR(A). All of the blocks, in R, of TZ(A), 1 mIm ⊗ trR(A) and the blocks of their exponentials commute. Using Theorem 2.1, Lemma 2.1 and the fact that trR(TZ(A)) = 0, we find KTZ(A) detR(eA) = KTZ(A) detR ( eTZ(A)+ 1 m Im⊗trR(A) ) = KTZ(A) detR ( eTZ(A)e 1 m Im⊗trR(A) ) = KTZ(A) detR ( eTZ(A) ) detR ( e 1 m Im⊗trR(A) ) = KTZ(A) detR ( e 1 m Im⊗trR(A) ) and by bilinearity of the Kronecker product and [9, Exercise 6.2.13], KTZ(A) detR(eA) = KTZ(A) detR ( Im ⊗ e 1 m trR(A) ) = KTZ(A)etrR(A). � 2888 A. FOŠNER ET AL. Corollary 2.1: Let F be the field of real numbers or the field of complex numbers. Let R be a commutative subring (with identity) of Mn(F) which is closed under scalar multiplication. Then for all A ∈ Mm(R), K detR(eA) = KetrR(A), (4) where K = { detR(eA) : A ∈ ker(trR) }. Moreover, K is the smallest subgroup of G satisfy- ing (4). Proof: Since, for each A ∈ Mn(R), KTZ(A) ⊆ K is a subgroup of K, we have K = KKTZ(A) and hence by Theorem 2.2, K detR(eA) = KKTZ(A) detR(eA) = KKTZ(A)etrR(A) = KetrR(A), so that (4) holds. Now let K ′ be a subgroup of G which satisfies (4) for all A ∈ Mn(R). Let A ∈ ker(trR), then K ′ detR(eA) = K ′etrR(A) = K ′ and since In ∈ K ′ it follows that detR(eA) ∈ K ′, i.e. K ⊆ K ′. � Of course, when R is the field of real numbers or the field of complex numbers, we have K = {1} and det(eA) = etr(A). 3. Partial determinants 3.1. Kronecker products Let In denote the n × n identity matrix, and 0n the n × n zero matrix. First we note that det1(Imn) = det1(Im ⊗ In) = In analogous to det(I) = 1. Definition 3.1: Let A,B ∈ Mn(R). The Hadamard product of A and B, A ◦ B, is the entry- wise product,⎛ ⎜⎝ a11 · · · a1n ... . . . ... an1 · · · ann ⎞ ⎟⎠ ◦ ⎛ ⎜⎝ b11 · · · b1n ... . . . ... bn1 · · · bnn ⎞ ⎟⎠ = ⎛ ⎜⎝ a11b11 · · · a11b1n ... . . . ... an1bn1 · · · annbnn ⎞ ⎟⎠ . Letm ∈ N. Them-thHadamard power ofA, denotedA(m) is thematrix defined inductively as the Hadamard product A(m) = A ◦ A(m−1), where A(1) := A. Lemma 3.1: Let A ∈ Mm(F) and B ∈ Mn(F). Then det1(A ⊗ B) = det(A)B(m). Proof: The proof follows immediately from A ⊗ B = n∑ i,j=1 ((B)ijA)⊗ E[n]ij , where B = ((B)ij). � LINEAR ANDMULTILINEAR ALGEBRA 2889 This is a direct consequence of the remark by Choi that det1(A)may be computed as a determinant where the blocks are treated as scalars and instead of the usual scalar product we use the Hadamard product [1]. Theorem 3.1: Let A,C ∈ Mm(F) and B,D ∈ Mn(F) with B ◦ D = 0n. Then det1(A ⊗ B + C ⊗ D) = det1(A ⊗ B)+ det1(C ⊗ D). Proof: We have A ⊗ B + C ⊗ D = n∑ i,j=1 (B)ij �=0 ((B)ijA)⊗ E[n]ij + n∑ i,j=1 (D)ij �=0 ((D)ijC)⊗ E[n]ij , (5) and since B ◦ D = 0n if follows that (B)ij �= 0 =⇒ (D)ij = 0 and vice-versa. The two sums in (5) are over disjoint subsets of {1, . . . , n}2 and det1(A ⊗ B + C ⊗ D) = n∑ i,j=1 (B)ij �=0 det(A)(B)mij E [n] ij + n∑ i,j=1 (D)ij �=0 det(C)(D)mij E [n] ij = det1(A ⊗ B)+ det1(C ⊗ D). � Lemma 3.1 provides an immediate characterization of partial determinants that can be completed. Theorem 3.2: Let A ∈ Mm(F) and B ∈ Mn(F). Then det(det1(A ⊗ B)) = det(A ⊗ B) if and only if det(A) = 0 or det(B(m)) = det(B)m. Drnovšek considered a much stronger condition in [15] for matrices B (which also satisfy Theorem 3.2), namely that B(r) = Br for all r ∈ N. Under the determinant, we have a weaker condition which leads to somewhat trivial cases for triangular matrices and (0, 1)-matrices. For triangular matrices B, we have det(B(m)) = det(Bm) = det(B)m. Corollary 3.1: Let A ∈ Mm(F) be arbitrary and let B ∈ Mn(F) be triangular. Then det(det1(A ⊗ B)) = det(A ⊗ B). For (0, 1)-matrices B we have B(m) = B, and hence the following corollary of Theorem 3.2. Corollary 3.2: Let A ∈ Mm(F) be arbitrary and let B ∈ Mn(F) be a (0, 1)-matrix. Then det(det1(A ⊗ B)) = det(A ⊗ B) if and only if det(A) det(B) = 0 or det(B) is an (m − 1)-th root of unity. 2890 A. FOŠNER ET AL. When n = 2, the characterization is straightforward since det ( a2 b2 c2 d2 ) = det ( a b c d )2 if and only if b2c2 = abcd, so that bc = 0 (triangular matrix) or ad = bc �= 0 (rank-1 matrix) and both cases satisfy det(B(m)) = det(B)m. Here we used the fact that a rank- 1 matrix xyT remains a rank-1 matrix x(m)(y(m))T under the Hadamard power, where x and y are column vectors. Now we consider transformations which preserve the partial determinant. The following theorem characterizes themultiplicative property of the partial determinant. Theorem 3.3: Let A,B ∈ Mm(F) and C,D ∈ Mn(F). Then det1((A ⊗ C)(B ⊗ D)) = det1(A ⊗ C) det1(B ⊗ D) if and only if det(AB) = 0 or (CD)(m) = C(m)D(m). Proof: Applying Lemma 3.1 on both sides of the equation provides det(AB)(CD)(m) = det(AB)C(m)D(m) and the result follows. � When F = GF(2), the Galois field of two elements, we have the following corollary. Corollary 3.3: Let A,B ∈ Mm(GF(2)) and C,D ∈ Mn(GF(2)). Then det1((A ⊗ C)(B ⊗ D)) = det1(A ⊗ C) det1(B ⊗ D). Since a row or column permutation of amatrix commutes with the Hadamard power, as does multiplying each row by a constant, we have the following corollary to Theorem 3.3. Corollary 3.4: Let A,B ∈ Mm(F) and C,P ∈ Mn(F), where P is a permutation matrix or a diagonal matrix. Then det1((AB)⊗ (PC)) = det1(A ⊗ P) det1(B ⊗ C), det1((AB)⊗ (CP)) = det1(A ⊗ C) det1(B ⊗ P). Thus, if B ∈ Mn(F) has no more than n(n + 1)/2 non-zero entries, and there exist per- mutations P,Q ∈ Mn(F) such that PBQT is triangular, then det(det1(A ⊗ B)) = det(A ⊗ B). Of course, this is not true in general. For example, B = ⎛ ⎜⎜⎝ 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 ⎞ ⎟⎟⎠ has 7 ≤ 10 non-zero entries, but no such P and Q exist. We note that multiplying by P preserves the number of non-zero entries in each column andmultiplying byQT preserves the number of non-zero entries in each row. LINEAR ANDMULTILINEAR ALGEBRA 2891 3.2. The exponential map Finally, when F = R or F = C, we have det(exp(A)) = exp(tr(A)) for all square matrices A. Theorem 3.4: Let B ∈ Mm(F) and C ∈ Mn(C) and let A be the Kronecker sumA = B ⊕ C. Then det1(exp(A)) = exp(tr1(A)) if and only if the m-th Hadamard and matrix powers of exp(C) coincide, i.e. (exp(C))(m) = (exp(C))m. The proof follows from exp(A) = exp(B)⊗ exp(C), Lemma 3.1 and the fact that det(exp(B)) = exp(tr(B)). 4. Determinant-roots The monoids of matrices described in Theorem 3.3 also satisfy Theorem 3.2. Let Hm ⊆ Mn(F) be a submonoid undermatrixmultiplication such thatA,B ∈ Hm satisfy (AB)(m) = A(m)B(m). Then Hm is a submonoid satisfying det(A(m)) = det(A)m for all A ∈ Hm. This follows since det(I(m)n ) = det(In)m, det(AB)m = det(A)m det(B)m = det(A(m)) det(B(m)) = det(A(m)B(m)) = det((AB)(m)). This monoid structure motivates the definition of a determinant-like operation which satisfies theorems analogous to Theorems 3.2 and 3.3. If the underlying field is alge- braically closed, then the analogous theorems hold inMn(F) (i.e. for all matrices). In this sectionwe definemonoids and a ‘determinant-root’ operationDet which is completable on Kronecker products of matrices in these monoids. First, let us define them-th roots on the multiplicative group F× of F. 4.1. Fields withm-th roots of unity Let G be a multiplicative abelian group, and Rm be the subgroup Rm = { g ∈ G : gm = 1 } (6) of G. The cosets of Rm are the equivalence classes a · Rm, where a · Rm = b · Rm if and only if am = bm (7) for a, b ∈ G. In particular, if G = F× – the multiplicative group in F – we obtain the following definition. Definition 4.1: Let F be a field and a, b ∈ F, and define Rm := { g ∈ F : gm = 1, g �= 0 }. We write a · Rm = m√b when am = b. If no such a ∈ F exists for a given b ∈ F, then m√b is undefined. Furthermore, we define m√0 := 0 · Rm. 2892 A. FOŠNER ET AL. We note that m√ab = m√a · m√b when the relevantm-th roots exist. The non-zerom-th roots of F× form a multiplicative group, and the matrices M ∈ Mn(F) where n√det(M) exists form a monoid Dn(F). In other words, since Mn(F) forms a monoid under matrix multiplication, Dn(F) ⊆ Mn(F) is a submonoid since In ∈ Dn(F) ( n√det(In) = 1 exists), and n√det(AB) = n√det(A) n√det(B) exists whenever n√det(A) and n√det(B) exist. Let m√a be them-th root of a ∈ F. Since m√a is in the (multiplicative) quotientF×/Rm, we consider the n-th root n √ m√a, when it exists. In this case, we may write mn√a ≡ n √ m√a, since G/Rmn(G) ∼= (G/Rm(G)) /Rn(G/Rm(G)), (8) where Rm(G) is the subgroup of the abelian group G as above. The isomorphism is simply aRmn(G) �→ (a · Rm(G)) · Rn(G/Rm(G)). We will elaborate more on this isomorphism in Lemma 4.2. In the following we develop a hierarchy of fields from repeated quotients of the form F×/Rm. This development holds for fields where every element has an m-th root (for example, when F = C is the field of complex numbers), but also holds for commutative division rigs (commutative semirings with multiplicative inverse) where every element has an m-th root (for example, the non- negative real numbers). Lemma 4.1: Let F denote a field with multiplication group G = F× such that every element has an m-th root and let Rm denotes the subgroup (6). Then Fm := F×/Rm ∪ {0 · Rm} is a field with operations (a · Rm) · (b · Rm) = (ab) · Rm, (a · Rm)+ (b · Rm) = m√am + bm · Rm. Proof: In the following, m√x denotes any m-th root of x ∈ F and the statements hold true for any m-th root. Then Rm is not empty since 1 ∈ Rm. Moreover, since a, b ∈ Rm we have (ab−1)m = am(bm)−1 = 1 since F× is commutative – i.e. ab−1 ∈ Rm. Clearly, Rm is a (commutative) normal subgroup of F×. Hence, F×/Rm is a commutative group with operation (a · Rm) · (b · Rm) = (ab) · Rm. We will show that the multiplicative group may be extended to a field by defining an appropriate additive structure. We note that a · Rm = c · Rm if and only if ac−1 ∈ Rm if and only if amc−m = 1 if and only if am = cm. Addition is well defined since a · Rm is uniquely determined by am. For example, if a · Rm = c · Rm and b · Rm = d · Rm, then a · Rm + b · Rm = m√am + bm · Rm = m√cm + dm · Rm = c · Rm + d · Rm. Clearly 0 · Rm is an identity element for addition. Now, additive inverses exist since −1 ∈ F× has anm-th root m√−1 and m√0 = 0 is the uniquem-th root of 0 ∈ F, a · Rm + ( m√−1a) · Rm = m √ am + ( m√−1a)m · Rm = m √ am + (−1)am · Rm = 0 · Rm. It remains to show that multiplicative distributivity holds. We have a m√x · Rm = m√amx · LINEAR ANDMULTILINEAR ALGEBRA 2893 Rm if and only if (a m√x)m = ( m√amx)m which holds true. Thus, (a · Rm)(b · Rm + c · Rm) = (a · Rm)( m√bm + cm · Rm) = (a m√bm + cm) · Rm = m √ (ab)m + (ac)m · Rm = (ab) · Rm + (ac) · Rm = (a · Rm) · (b · Rm)+ (a · Rm) · (c · Rm). We also note that F1 ≡ F. � Lemma 4.2: Let F denote a field such that every element has an m-th root and an mn-th root. Then Fmn ∼= (Fm)n. Here Fm = F×/Rm ∪ {0 · Rm} with operations as in Lemma 4.1 and similarly for Fmn. Proof: First we must show that every element of Fm has an n-th root. Let a ∈ F. Since a has mn-th roots there exists b ∈ F such that bmn = a, and, hence, bm is an n-th root of a and bm · Rm is an n-th root of a · Rm. It remains to demonstrate that there exists a field isomorphism φ : Fmn → (Fm)n. Define φ(a · Rmn) = (a · Rm) · Rn, where Rm = Rm(F) is the multiplicative subgroup (6) of F and Rn = Rn(Fm) is the mul- tiplicative subgroup (6) of Fm. The function φ is well-defined since for all x ∈ a · Rmn, (xn)m = (an)m so that xn · Rm = an · Rm and, finally, (x · Rm) · Rn = (a · Rm) · Rn. Injec- tivity of φ follows similarly and surjectivity is immediate. Finally, φ is a ring homomor- phism since φ((a · Rmn) · (b · Rmn)) = [(a · Rm) · (b · Rm)] · Rn = [(a · Rm) · Rn] · [(b · Rm) · Rn] = φ(a · Rmn) · φ(b · Rmn), φ(a · Rmn + b · Rmn) = φ( mn√amn + bmn · Rmn) = ( mn√amn + bmn · Rm) · Rn = n √ an · Rm + bn · Rm · Rn = (a · Rm) · Rn + (b · Rm) · Rn = φ(a · Rmn)+ φ(b · Rmn). � 2894 A. FOŠNER ET AL. There is a natural embedding hm,n : a · Rm �→ an · Rmn which is well defined since if a · Rm = b · Rm then am = bm and hence, (an)mn = (bn)mn, which yields that hm,n(a · Rm) = hm,n(b · Rm). We define the product ∗ as follows (a · Rm) ∗ (b · Rn) := hm,n(a · Rm) · hn,m(b · Rn) = (anbm) · Rmn. (9) Consequently, m√a ∗ n√b = mn√anbm when the given roots exist. Lemma 4.3: Let F denote a field such that every element has an m-th root and an mn-th root. Then hm,n : Fm → Fmn is a (multiplicative) group homomorphism. Proof: The proof follows straightforwardly from the definitions hm,n((a · Rm) · (b · Rm)) = (ab)n · Rmn = anbn · Rmn = (an · Rmn) · (bn · Rmn) = hm,n(a · Rm) · hm,n(b · Rm). � The fields Fm are defined when all elements have m-th roots. Hence, it is necessary to consider a theory over the group (or binoid, if 0 is included) F×/Rm and additional results in Fm when it exists. 4.2. The determinant-root on Dn(F) Recall that Dn(F) denotes the matrices A ∈ Mn(F) such that det(A) has an n-th root in F. Definition 4.2: We define the determinant-root Det by Det(M) := n √ det(M) for allM ∈ Dn(F). Remark 4.1: When n = 2 and F = R, Det(M) is equivalent to the non-negative square root of det(M). This definition provides the usual multiplicative property Det(AB) = Det(A)Det(B) for A,B ∈ Dn(F). With respect to scalar multiplication however, we have Det(αA) = n √ det(αA) = (α · Rn) n √ det(A) = (α · Rn)Det(A). By the embedding (9), we have Det(A ⊗ C) = Det(A) ∗ Det(C) for A ∈ Dm(F) and C ∈ Dn(F) from det(A ⊗ C) = (det(A))n(det(C))m. LINEAR ANDMULTILINEAR ALGEBRA 2895 Nowwe are ready to consider partial determinant-roots and their properties. The partial determinant-root Det2 is again the block-wise determinant-root. Definition 4.3: Let A,B ∈ Mmn(F) = Mm(Mn(F)) = Mm(F)⊗ Mn(F) with A = n∑ i,j=1 Aij ⊗ E[n]ij , B = m∑ i,j=1 E[m] ij ⊗ Bij and assume that Aij ∈ Dm(F) and Bij ∈ Dn(F) for all i, j. The partial determinant-roots Det1(A) and Det2(B) are given by Det1(A) := n∑ i,j=1 Det(Aij)E [n] ij , Det2(B) := m∑ i,j=1 Det(Bij)E [m] ij . In general, we work in the group ring Z[F×/Rn(F×)], i.e. Det2 : Mm(Dn(F)) → Mm(Z[F×/Rm(F×)]) and similarly for Det1. However, when we are concerned only with Kronecker products of matrices, we will express our results inF×/Rm(F×)where possible. Additionally, when Fn exists, we assume Det2 : Mm(F)⊗ Mn(F) → Mm(Fn). Lemma 4.4: The partial determinant-root obeys Det1(A ⊗ B) = Det(A)B, where Det(A)B is the entry-wise product of the entries in B with Det(A). Proof: We have A ⊗ B = n∑ i,j=1 ((B)ijA)⊗ E[n]ij , where B = (B)ij. It follows that Det1(A ⊗ B) = n∑ i,j=1 Det(BijA)E [n] ij = n∑ i,j=1 BijDet(A)E [n] ij = Det(A) n∑ i,j=1 BijE [n] ij = Det(A)B, since m √ Bmij = Bij · Rm(F×). � Theorem 4.1: Let A ∈ Dm(F) and C ∈ Dn(F). Then Det(A ⊗ C) = Det(Det1(A ⊗ C)). Proof: Since Det(A ⊗ C) = Det(A) ∗ Det(C) and (by the isomorphism (8)) Det(Det1(A ⊗ C)) = n √ det(Det(A)C) = n √ (Det(A))n det(C) 2896 A. FOŠNER ET AL. ≡ mn √ (det(A))n(det(C))m = Det(A) ∗ Det(C) we obtain Det(A ⊗ C) = Det(Det1(A ⊗ C)). � Lemma 4.5: Let F denote a field such that every element has an m-th root. Then Det(A) = Det(Det1(A)). for all A ∈ Mmn(F) if and only if Det(Det1(XY)) = Det(Det1(X)Det1(Y)) for all X,Y ∈ Mmn(F) and similarly for Det2. Proof: (=⇒) It follows by direct calculation Det(Det1(XY)) = Det(XY) = Det(X)Det(Y) = Det(Det1(X))Det(Det1(Y)) = Det(Det1(X)Det1(Y)). (⇐ = ) The proof is straightforward when X = A and Y = Imn, � Lemma 4.6: Let F denote a field such that every element has an mn-th root. Then Det(A) = Det(Det2(A)) for all A ∈ Mmn(F) if and only if Det(X + Y) = Det(X)+ Det(Y) in (Fn)m for all X,Y ∈ Mn(F). Proof: (=⇒) Assume that Det(A) = Det(Det2(A)) for allA ∈ Mmn(F). We will show that m √ Det(X + Y) = m √ Det(X)+ Det(Y) in (Fn)m for all X,Y ∈ Mn(F). By uniqueness ofm-th roots in (Fn)m we will conclude that Det(X + Y) = Det(X)+ Det(Y) in (Fn)m. Using Definition 4.1 and the isomorphism in Lemma 4.2, we have m √ Det(X + Y) = Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ X + Y In 0 · · · 0 0 In 0 · · · 0 0 0 In . . . 0 ... ... . . . . . . 0 0 0 0 · · · In ⎞ ⎟⎟⎟⎟⎟⎟⎠ LINEAR ANDMULTILINEAR ALGEBRA 2897 = Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ X In 0 · · · 0 −Y In 0 · · · 0 0 0 In . . . 0 ... ... . . . . . . 0 0 0 0 · · · In ⎞ ⎟⎟⎟⎟⎟⎟⎠ Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ In 0 0 · · · 0 Y In 0 · · · 0 0 0 In . . . 0 ... ... . . . . . . 0 0 0 0 · · · In ⎞ ⎟⎟⎟⎟⎟⎟⎠ and, by assumption Det(A) = Det(Det2(A)), m √ Det(X + Y) = Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ Det2 ⎛ ⎜⎜⎜⎜⎜⎜⎝ X In 0 · · · 0 −Y In 0 · · · 0 0 0 In . . . 0 ... ... . . . . . . 0 0 0 0 · · · In ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎠ Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ Det2 ⎛ ⎜⎜⎜⎜⎜⎜⎝ In 0 0 · · · 0 Y In 0 · · · 0 0 0 In . . . 0 ... ... . . . . . . 0 0 0 0 · · · In ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎠ = Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ Det(X) 1 0 · · · 0 −Det(Y) 1 0 · · · 0 0 0 1 . . . 0 ... ... . . . . . . 0 0 0 0 · · · 1 ⎞ ⎟⎟⎟⎟⎟⎟⎠ Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ 1 0 0 · · · 0 Det(Y) 1 0 · · · 0 0 0 1 . . . 0 ... ... . . . . . . 0 0 0 0 · · · 1 ⎞ ⎟⎟⎟⎟⎟⎟⎠ = Det ⎛ ⎜⎜⎜⎜⎜⎜⎝ Det(X)+ Det(Y) 1 0 · · · 0 0 1 0 · · · 0 0 0 1 . . . 0 ... ... . . . . . . 0 0 0 0 · · · 1 ⎞ ⎟⎟⎟⎟⎟⎟⎠ = m √ Det(X)+ Det(Y). (⇐=) If Det(X + Y) = Det(X)+ Det(Y) in (Fn)m, then Det is linear in the sense that Det(αA) = (α · Rm)Det(A) for all α ∈ Fn, Using the isomorphism in Lemma 4.2, Det(A) = Det ⎛ ⎝ m∑ i,j=1 Eij ⊗ A′ ij ⎞ ⎠ = m∑ i,j=1 Det ( Eij ⊗ A′ ij ) = m∑ i,j=1 Det(Eij) ∗ Det(A′ ij) = m∑ i,j=1 mn √ det(Eij)n det(A′ ij) m · Rmn = m∑ i,j=1 mn √( det(Eij) n √ det(A′ ij) m)n · Rmn = m∑ i,j=1 mn √ det ( n √ det(A′ ij)Eij )n · Rmn � m∑ i,j=1 m √ det(Det(A′ ij)Eij) · Rm = m∑ i,j=1 Det(Det(A′ ij)Eij) 2898 A. FOŠNER ET AL. = Det ⎛ ⎝ m∑ i,j=1 Det(A′ ij)Eij ⎞ ⎠ = Det(Det2(A)). � Corollary 4.1: If Det(A) = Det(Det2(A)) for all A ∈ Mmn(F), then Det(A) = 0 for all A ∈ Mmn(F). Proof: We have, by Lemma 4.6, Det(A) = Det ⎛ ⎝ m∑ i,j=1 Eij ⊗ A′ ij ⎞ ⎠ = m∑ i,j=1 Det ( Eij ⊗ A′ ij ) = m∑ i,j=1 Det(Eij) ∗ Det(A′ ij) = 0. � Corollary 4.2: Theorem 4.1 can only be extended to all matrices in a trivial sense, i.e. Det(A) ≡ 0. 4.3. Products Next we show that partial determinant-roots are multiplicative on products of Kronecker products. Theorem 4.2: Let A,B ∈ Dm(F) and C,D ∈ Mn(F). Then Det1((A ⊗ C)(B ⊗ D)) = Det1(A ⊗ C)Det1(B ⊗ D). Proof: Lemma 4.4 provides Det1((A ⊗ C)(B ⊗ D)) = Det(AB) (CD). Similarly, Det1(A ⊗ C)Det1(B ⊗ D) = (Det(A)C)(Det(B)D) = (Det(A)Det(B)) (CD) = Det(AB) (CD). � LINEAR ANDMULTILINEAR ALGEBRA 2899 Lemma 4.7: Let F denote a field such that every element has an n-th root. Then Det2(AB) = Det2(A)Det2(B). for all A,B ∈ Mmn(F) if and only if Det(X + Y) = Det(X)+ Det(Y) for all X,Y ∈ Mn(F) and similarly for Det1. Proof: (=⇒) Assume that Det2(AB) = Det2(A)Det2(B) for all A,B ∈ Mmn(F). In partic- ular, for all X,Y ∈ Mn(F),⎛ ⎜⎜⎜⎝ Det(X)+ Det(Y) 0 · · · 0 0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎝ Det(X) 1 · · · 1 0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎝ 1 0 · · · 0 Det(Y) 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ = Det2 ⎛ ⎜⎜⎜⎝ X In · · · In 0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠Det2 ⎛ ⎜⎜⎜⎝ In 0 · · · 0 Y 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ = Det2 ⎛ ⎜⎜⎜⎝ X + Y 0 · · · 0 0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎝ Det(X + Y) 0 · · · 0 0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 ⎞ ⎟⎟⎟⎠ , and, consequently, Det(X + Y) = Det(X)+ Det(Y). (=⇒) Suppose that Det(X + Y) = Det(X)+ Det(Y) for all X,Y ∈ Mn(F). By direct cal- culation, Det2(AB) = Det2 ⎛ ⎝ m∑ i,j=1 Eij ⊗ m∑ k=1 A′ ikB ′ kj ⎞ ⎠ = m∑ i,j=1 Det ( m∑ k=1 A′ ikB ′ kj ) Eij = m∑ i,j,k=1 Det ( A′ ikB ′ kj ) Eij = m∑ i,j,k=1 Det(A′ ik)Det(B ′ kj)Eij = ⎛ ⎝ m∑ i,k=1 Det(A′ ik)Eik ⎞ ⎠ ⎛ ⎝ m∑ j,k=1 Det(B′ kj)Ekj ⎞ ⎠ = Det2(A)Det2(B). � 2900 A. FOŠNER ET AL. Corollary 4.3: Theorem 4.2 can only be extended to all matrices in a trivial sense, i.e. Det(A) ≡ 0 since, Det(A) = Det ⎛ ⎝ m∑ i,j=1 aijEij ⎞ ⎠ = m∑ i,j=1 Det(aijEij) = 0. 4.4. The exponential map Assume that n �= 0 in F. Define Tr : Mn(F) → F by Tr : A �→ tr(A) n and define the partial operation Tr1 in the natural way. For determinant-roots we obtain Det(exp(A)) = exp(Tr(A)) · Rn(F×). (10) Partial determinant-roots obey the exponential-determinant-trace relation as follows. Theorem 4.3: Let F be the field F = R of real numbers or the field F = C of complex numbers and let A be the Kronecker sum A = B ⊕ C. Then Det1(exp(A)) = exp(Tr1(A)) · Rm(F×). Proof: The result follows from exp(Tr1(A)) = exp(1/m tr(B)In + C) = e1/m tr(B)eC and, since eB ∈ DM(F) (for F = R or F = C) and using Lemma 4.4, Det1(exp(A)) = Det1(eB ⊗ eC) = Det(eB)eC = m √ det(eB) eC = m √ etr(B) eC = e1/m tr(B)eC · Rm(F×). � 5. Non-linear preservers of determinants In this section we discuss a class of preservers of determinants, i.e. maps ψ : Mn(F) → Mn(F)which satisfy det(ψ(A)) = det(A). In particular, we studymaps whichmay bewrit- ten in the form ψ(eA) = eφ(A), where φ : Mn(F) → Mn(F) is a linear map as described in (11) below. Naturally, this implies a restriction on the matricesA and we will find it con- venient to reframe the problem in terms of the determinant-root Det from Definition 4.2. LINEAR ANDMULTILINEAR ALGEBRA 2901 In this setting, we may study new classes of maps satisfying Det(ψ(A ⊗ B)) = Det(A ⊗ B), i.e. we will continue the study of [11] in this new setting. Since the determinant, trace and Kronecker product are related by the equations det(eA) = etr(A), eA ⊗ eB = eA⊕B, over the field R or C, we will first study the Kronecker sum ⊕ and its linear preservers. 5.1. RT-Symmetry Noting that Mn(F) is an n2-dimensional space, for an arbitrary linear map φ : Mn(F) → Mn(F), we have φ(A) = n∑ j,k,u,v=1 αjk;uvE [n] jk AE[n]uv (11) for some αjk;uv ∈ F in the underlying field. Definition 5.1: LetF be an arbitrary field and φ : Mn(F) → Mn(F) be a linear map in the form (11). We define the linear transform ′ : φ �→ φ′, by φ′(A) := n∑ j,k,u,v=1 αuv;jkE [n] jk AE[n]uv ≡ n∑ j,k,u,v=1 αjk;uvE[n]uv AE [n] jk . (12) When F = C is the field of complex numbers, we denote by φ∗ := φ′ the complex conjugate composed with φ′, i.e. φ∗(A) = φ′(A). The following two lemmas follow immediately. Lemma 5.1: Let φ,ψ : Mn(F) → Mn(F) be linear maps. Then (a) (φ′)′ = φ, (b) (αφ + ψ)′ = α(φ′)+ ψ ′ for all α ∈ F. Lemma 5.2: Let φ,ψ : Mn(C) → Mn(C) be linear maps. Then (a) (φ∗)∗ = φ, (b) (φ + ψ)∗ = φ∗ + ψ∗. In the following, we will associate to every map φ a matrix�, which is defined in terms of the vec operator. Given amatrixA, let vec(A) be the column vector obtained by stacking the columns ofA on top of each other [7]. The following lemma shows that the vec operator maps rank-one matrices to Kronecker products of column vectors. Lemma 5.3: Let a, b ∈ Fn be column vectors with entries from the field F. Then vec(abT) = b ⊗ a. 2902 A. FOŠNER ET AL. Proof: Straightforward calculation yields that vec(abT) = vec ⎛ ⎜⎜⎜⎝ a1b1 a1b2 · · · a1bn a2b1 a2b2 · · · a2bn ... ... . . . ... anb1 anb2 · · · anbn ⎞ ⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ a1b1 a2b1 ... anb1 ... ... a1bn a2bn ... anbn ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = b ⊗ a. � Let� be the matrix representing the linear map φ in the standard basis, i.e.� vec(A) = vec(φ(A)). Similarly, let�′ be the matrix representing φ′. Since φ(E[n]pq ) = n∑ j,v=1 αjp;qvE [n] jv , φ′(E[n]pq ) = n∑ j,v=1 αqv;jpE [n] jv , it follows by Lemma 5.3 that � = n∑ j,p,q,v=1 αjp;qvE[n]vq ⊗ E[n]jp , �′ = n∑ j,p,q,v=1 αqv;jpE[n]vq ⊗ E[n]jp . Definition 5.2 (Perfect shuffle matrix [7,8]): The perfect shuffle matrix P is the unique permutation matrix which satisfies PT(A ⊗ B)P = B ⊗ A for every A,B ∈ Mn(R), i.e. P = n∑ j=1 n∑ k=1 E[n]jk ⊗ E[n]kj . The perfect shuffle matrix is also known as the vec-permutation matrix on Fn ⊗ Fn. It follows that �T = PT�′P. (13) Definition 5.3 (Rearrangement operator [16]): The rearrangement operator R is linear and defined by R(A ⊗ B) = (vecA)(vecB)T , where vec is the vec operator [7]. The rearrangement operator provides the characterization of φ which obeys φ = φ′. Lemma 5.4: Let φ : Mn(F) → Mn(F) be a linear map and� be the matrix satisfying � vec(A) = vec(φ(A)). Then the following are equivalent: LINEAR ANDMULTILINEAR ALGEBRA 2903 (a) φ = φ′, (b) �T = PT�P, (c) R(�)T = R(�T), where P is the perfect shuffle matrix (Definition 5.2) and R is the rearrangement operator (Definition 5.3). Proof: The proof (a) ⇐⇒ (b) follows since � is uniquely determined by PT(A ⊗ B)P = B ⊗ A, and by (13). By simple calculation, if φ = φ′, then using the forms of φ in (11) and of φ′ in (12) we have that αjp;qv = αqv;jp, R(�T) = n∑ j,p,q,v=1 αjp;qvR(E [n] pj ⊗ E[n]qv ) = n∑ j,p,q,v=1 αjp;qv vec(E [n] pj ) vec(E [n] qv ) T = n∑ j,p,q,v=1 αjp;qvE [n] jv ⊗ E[n]pq = n∑ j,p,q,v=1 αjp;qvE [n] qp ⊗ E[n]vj , R(�) = n∑ j,p,q,v=1 αjp;qvR(E [n] jp ⊗ E[n]vq ) = n∑ j,p,q,v=1 αjp;qv vec(E [n] jp ) vec(E [n] vq ) T = n∑ j,p,q,v=1 αjp;qvE [n] pq ⊗ E[n]jv = R(�T)T . On the other hand, comparing coefficients ofE[n]pq ⊗ E[n]jv inR(�)T = R(�T) yieldsαjp;qv = αqv;jp so that φ = φ′. We have proved (a) ⇐⇒ (c). � Lemma 5.4(c) motivates the following definitions. Definition 5.4: A linear map φ : Mn(F) → Mn(F) satisfying φ = φ′ is said to be RT- symmetric. If φ = −φ′ then φ is said to be skew RT-symmetric. Definition 5.5: A linear map φ : Mn(C) → Mn(C) satisfying φ = φ′ is said to be RT- Hermitian. If φ = −φ′ then φ is said to be skew RT-Hermitian. The following lemma follows immediately from φ = 1 2 (φ + φ′)+ 1 2 (φ − φ′). Lemma 5.5: If the underlying field has characteristic not equal to 2, then every linear map φ : Mn(F) → Mn(F) is the sum of an RT-symmetric map and a skew RT-symmetric map. 5.2. Linear trace preservers of Kronecker sums Wemay write a linear map φ : Mmn → Mmn in the operator-sum form φ(M) = r∑ i=1 PiMQi 2904 A. FOŠNER ET AL. for some matrices Pi,Qi, i = 1, . . . , r, of the appropriate sizes. If φ preserves the trace of a Kronecker sumM, then the cyclic property of the trace yields tr(M) = tr(φ(M)) = r∑ i=1 tr(QiPiM) and so we need only consider preservers of the form φ(M) = r∑ i=1 QiPiM = PM, where P := r∑ i=1 QiPi (14) and the remaining preservers are all obtained by representations (14) ofP. First we consider maps of the form φ(M) = PM,M ∈ Mmn. Theorem 5.1: Let φ : Mmn → Mmn be a map given by φ : M �→ PM for some P ∈ Mmn. Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr1(P) = tr1(Imn) and tr2(P) = tr2(Imn). Proof: First, let us write P in block matrix form, P = (P′ kl) where each P′ kl ∈ Mn(F) for k, l = 1, . . . ,m. In other words, P = m∑ k,l=1 E[m] kl ⊗ P′ kl. Since φ is linear, themap φ preserves the trace of Kronecker sums if and only if φ preserves traces of Kronecker products of the form E[m] ij ⊕ 0 = E[m] ij ⊗ In and of the form 0 ⊕ E[n]kl = Im ⊗ E[n]kl . Thus, we have nδij = tr(E[m] ij ⊗ In) = tr(φ(E[m] ij ⊗ In)) (15) and tr(φ(E[m] ij ⊗ In)) = m∑ k,l=1 tr((E[m] kl E[m] ij )⊗ P′ kl) = m∑ k,l=1 δil tr(E [m] kj ) tr(P ′ kl) = m∑ k,l=1 δjkδil tr(P′ kl) = tr(P′ ji). (16) LINEAR ANDMULTILINEAR ALGEBRA 2905 It follows, by comparing (15) and (16), that tr2(P) = m∑ k,l=1 tr(P′ kl)E [m] kl = m∑ k,l=1 nδklE [m] kl = nIm. For Kronecker products of the form Im ⊗ E[n]kl , we find mδkl = tr(Im ⊗ E[n]kl ) = tr(φ(Im ⊗ E[n]kl )), and since tr(PijE [n] kl ) = (P′ jj)lk, mδkl = tr(φ(Im ⊗ E[n]kl )) = m∑ i,j=1 tr(E[m] ij ⊗ (P′ ijE [n] kl )) = m∑ i,j=1 δij tr(P′ ijE [n] kl ) = m∑ j=1 (P′ jj)lk. Consequently, tr1(P) = m∑ j=1 P′ jj = mIn. Conversely, suppose that tr1(P) = mIn and tr2(P) = nIm. Then, by Corollary 1.1, tr(φ(A ⊕ B)) = tr(tr2(P(A ⊗ In)))+ tr(tr1(P(Im ⊗ B))) = tr(tr2(P)A)+ tr(tr1(P)B) = n tr(A)+ m tr(B) = tr(A ⊕ B). � Corollary 5.1: Let φ : Mmn → Mmn be a map given by φ : M �→ PM for some P ∈ Mmn, where P = Imn + r∑ j=1 Aj ⊗ Bj. Here, r is the tensor rank of P − Imn over Mm(F)⊗ Mn(F) and Aj ∈ Mm(F), Bj ∈ Mn(F) for j = 1, . . . , r. Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr(Aj) = tr(Bj) = 0 for j = 1, . . . , r. Proof: By Theorem 5.1, we only need to show that tr1(P) = mIn and tr2(P) = nIm if and only if tr(Aj) = tr(Bj) = 0 for j = 1, . . . , r. The proof of (⇐) is immediate. For (⇒), suppose tr1(P) = mIn and tr2(P) = nIm. It follows that tr1(Imn) = mIn = tr1(P) = tr1(Imn)+ r∑ j=1 tr(Aj)Bj ⇐⇒ r∑ j=1 tr(Aj)Bj = 0n, tr2(Imn) = nIm = tr2(P) = tr2(Imn)+ r∑ j=1 tr(Bj)Aj ⇐⇒ r∑ j=1 tr(Bj)Aj = 0m. Since r is the tensor rank of P − Imn, the set {B1, . . . , Br} is a linearly independent set and tr(Aj) = 0 for j = 1, . . . , r. Similarly, tr(Bj) = 0 for j = 1, . . . , r. � 2906 A. FOŠNER ET AL. As a consequence of Theorem 5.1, we have that φ : M �→ PM satisfies tr(φ(A ⊕ B)) = tr(A ⊕ B) if and only if tr1(φ(Imn)) = tr1(P) = tr1(Imn) and tr2(φ(Imn)) = tr2(P) = tr2(Imn). In general, this statement is true modulo a traceless matrix. We note that any linear map φ : Mmn → Mmn can be written in the form φ(M) = M + s∑ j=1 (Aj ⊗ Cj)M(Bj ⊗ Dj), where Aj,Bj ∈ Mm(F) and Cj,Dj ∈ Mn(F) for j = 1, . . . , s. In the following we denote by [A,B] = AB − BA the Lie product (the commutator of matricesA and B of the appropriate sizes). Lemma 5.6: Let φ : Mmn → Mmn be a linear map given by φ(M) = M + s∑ j=1 (Aj ⊗ Cj)M(Bj ⊗ Dj). Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr1(φ(Imn)− Imn) = s∑ j=1 tr(AjBj)[Cj,Dj] and tr2(φ(Imn)− Imn) = s∑ j=1 tr(CjDj)[Aj,Bj]. Proof: Since tr(φ(A ⊕ B)) = tr(A ⊕ B) if and only if tr(φ(A ⊗ In)) = tr(A ⊗ In) and tr(φ(Im ⊗ B)) = tr(Im ⊗ B) for all A ∈ Mm(F) and B ∈ Mn(F), we consider these two cases separately. In the first case we have by the cyclic property of the trace, tr(φ(A ⊗ In)) = tr(A ⊗ In)+ s∑ j=1 tr((Aj ⊗ Cj)(A ⊗ In)(Bj ⊗ Dj)) = n tr(A)+ s∑ j=1 tr ( ((BjAj)⊗ (DjCj))(A ⊗ In) ) = n tr(A)+ s∑ j=1 tr ( tr2(((BjAj)⊗ (DjCj))(A ⊗ In)) ) = n tr(A)+ s∑ j=1 tr ( tr2((BjAj)⊗ (DjCj))A ) (Corollary 1.1) LINEAR ANDMULTILINEAR ALGEBRA 2907 = n tr(A)+ tr ⎛ ⎝ ⎛ ⎝ s∑ j=1 tr(CjDj)BjAj ⎞ ⎠A ⎞ ⎠ (Proposition 1.1) for all A ∈ Mm(F). Thus, n tr(A) = tr(A ⊗ In) = tr(φ(A ⊗ In)) holds for all A if and only if 0m = s∑ j=1 tr(CjDj)BjAj = s∑ j=1 tr(CjDj)([Bj,Aj] + AjBj). (17) We note that tr2(φ(Imn))− tr2(Imn) = s∑ j=1 tr2((AjBj)⊗ (CjDj)) = s∑ j=1 tr(CjDj)AjBj. Defining the traceless matrix Q (i.e. tr(Q) = 0) by Q := s∑ j=1 tr(CjDj)[Aj,Bj] provides the condition equivalent to (17) 0m = −Q + s∑ j=1 tr(CjDj)AjBj = −Q + tr2(φ(Imn))− tr2(Imn). Hence, tr(A ⊗ In) = tr(φ(A ⊗ In)) for all A if and only if tr2(φ(Imn)− Imn) = s∑ j=1 tr(CjDj)[Aj,Bj]. Similarly, the second case yields that tr(φ(Im ⊗ B)) = tr(Im ⊗ B) for all B if and only if s∑ j=1 tr(AjBj)[Cj,Dj] = tr1(φ(Imn))− tr1(Imn). � The commutators in the above lemma highlight the traceless character [17]. Moreover, Lemma 5.6 shows that the partial traces of the identity matrix must be preserved modulo a traceless matrix. However, this traceless matrix is not arbitrary but precisely defined in terms of φ. The following theorem shows that φ′, defined as in Equation (12), plays a fundamental role in the characterization of φ and provides a suc- cinct characterization for RT-symmetric and skew RT-symmetric maps in the subsequent two corollaries. Theorem 5.2: Let φ : Mmn → Mmn be a linear map. Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr1(φ′(Imn)) = tr1(Imn) and tr2(φ′(Imn)) = tr2(Imn). 2908 A. FOŠNER ET AL. Proof: Using the representation of φ from Lemma 5.6 provides tr1(φ(Imn)) = tr1(Imn)+ s∑ j=1 tr(AjBj)CjDj, tr1(φ′(Imn)) = tr1(Imn)+ s∑ j=1 tr(BjAj)DjCj and subtracting these two equations yields tr1((φ − φ′)(Imn)) = s∑ j=1 tr(AjBj)[Cj,Dj]. Similarly, tr2((φ − φ′)(Imn)) = s∑ j=1 tr(CjDj)[Aj,Bj]. FromLemma5.6, tr(φ(A ⊕ B)) = tr(A ⊕ B) for allA ∈ Mm(F) andB ∈ Mn(F) if and only if tr1(φ(Imn)− Imn) = tr1((φ − φ′)(Imn)) and tr2(φ(Imn)− Imn) = tr2((φ − φ′)(Imn)) if and only if tr1(φ′(Imn)) = tr1(Imn) and tr2(φ′(Imn)) = tr2(Imn). � Corollary 5.2: Let φ : Mmn → Mmn be an RT-symmetric map. Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr1(φ(Imn)) = tr1(Imn) and tr2(φ(Imn)) = tr2(Imn). Corollary 5.3: Let φ : Mmn → Mmn be a skew RT-symmetric map. Then tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ∈ Mm(F) and B ∈ Mn(F) if and only if tr1(φ(Imn)) = − tr1(Imn) and tr2(φ(Imn)) = − tr2(Imn). It is straightforward to extend Corollaries 5.2 and 5.3 to the RT-Hermitian and skew RT-Hermitian cases since tr1(φ′(Imn)) = tr1(Imn) if and only if tr1(φ′(Imn)) = tr1(Imn). Now we are ready to consider the connection with the work in [11]. The connection is provided by the exponential map, i.e. det(eA ⊗ eB) = etr(A⊕B). LINEAR ANDMULTILINEAR ALGEBRA 2909 5.3. Determinant preservers of Kronecker products Denote by GLn(C) the general linear group of n × nmatrices over C with non-zero deter- minant and by SLn(C) the group of matrices with determinant equal to 1. In the following we will follow the notational convention Rn := Rn(C), i.e. Rn is the zero of the field Cn of Lemma 4.1. The condition given in Lemma 5.6 implies that we may characterize a class of determinant preservers of Kronecker products in terms of partial determinants. However, the relationship between the partial trace and the partial determinant is not straightfor- ward. If we restrict our attention to matrices over the complex numbers,Mmn(C), then we have from (10) Det(eA ⊗ eB) = eTr(A⊕B) · Rmn, where Det(A) := n√det(A)Rn and Tr(A) := tr(A)/n for A ∈ Mn(C), and Rn is the mul- tiplicative group of n-th roots of unity in C. Furthermore, we showed in Theorem 4.3 that Det1(eA ⊗ eB) = eTr1(A⊕B) · Rm and Det2(eA ⊗ eB) = eTr2(A⊕B) · Rn. Definition 5.6: Let n ⊂ Mn(C) denote the set of matrices A ∈ Mn(C) with the imagi- nary part of each eigenvalue λ(A) satisfying Im(λ(A)) ∈ (−π ,π]. Thus, we associate with every non-singular matrixA a uniquematrixM ∈ n such that A = eM [9, p. 474]. Next, we wish to consider linear maps restricted to n. The following lemma shows that trace preservers on n are equivalent to trace preservers onMn(C). Lemma 5.7: Let φ : Mn(C) → Mn(C) be a linear map obeying φ( n) ⊆ n. Then φ satisfies tr(φ(A)) = tr(A) for all A ∈ Mn(C) if and only if tr(φ(B)) = tr(B) for all B ∈ n. Proof: It suffices to show that each standard basis element E[n]ij ∈ Mn(C) is also in n, which is easily verified. � Let φ : Mn(C) → Mn(C) be a linear map obeying φ( n) ⊆ n and letψ : GLn(C) → GLn(C) be the non-linear map ψ(eM) = eφ(M), M ∈ n. The map is well defined sinceM ∈ n is uniquely determined for every matrix inGLn(C). We have detψ(eM) = det eφ(M) = etr(φ(M)) so that detψ(eM) = det(eM) if and only if etr(φ(M)) = etr(M). Moreover, if φ(M) ∈ n and M ∈ n, then eTr(φ(M)) = eTr(M) if and only if tr(φ(M)) = tr(M). The following theorem motivates the remainder of the study. 2910 A. FOŠNER ET AL. Theorem 5.3: Let φ : Mn(C) → Mn(C) be a linear map obeying φ( n) ⊆ n. The map ψ : GLn(C) → GLn(C) given by ψ(eM) = eφ(M), M ∈ n, satisfies det(ψ(eM/n)) = det(eM/n) if and only if tr(φ(M)) = tr(M). Proof: LetM ∈ n, thenφ(M) ∈ n so that Im(Tr(φ(M))), Im(Tr(M)) ∈ (−π ,π]. Thus, exp(Tr(φ(M))) = exp(Tr(M)) if and only if Tr(φ(M)) = Tr(M). Hence, tr(φ(M)) = tr(M) ⇐⇒ exp(Tr(φ(M))) = exp(Tr(M)) ⇐⇒ exp(tr(φ(M/n))) = exp(tr(M/n)) ⇐⇒ det(exp(φ(M/n)) = det(exp(M/n)) ⇐⇒ det(ψ(exp(M/n))) = det(exp(M/n)). � The condition det(ψ(eM/n)) = det(eM/n) implies (det(ψ(eM/n))n = (det(eM/n))n. Furthermore, from det(eM/n)n = det(eM) we have, by the uniqueness of n-th roots in Rn, that Det(eM) = det(eM/n) · Rn. Similarly, by linearity of φ, we have Det(ψ(eM)) = det(ψ(eM/n)) · Rn. Thus, Det(ψ(eM)) = det(ψ(eM/n)) · Rn = det(eM/n) · Rn = Det(eM), for allM ∈ n. Hence, the following corollary. Corollary 5.4: Let φ : Mn(C) → Mn(C) be a linear map such that φ( n) ⊆ n. If tr(φ(M)) = tr(M) for all M ∈ n, then the map ψ : GLn(C) → GLn(C) given by ψ(eM) = eφ(M), M ∈ n, satisfies Det(ψ(eM)) = Det(eM) for all M ∈ n. Lemma 5.8: Let φ : Mn(C) → Mn(C) be a linear map such that φ( n) ⊆ n. Then the map ψ : GLn(C) → GLn(C) given by ψ(eM) = eφ(M), M ∈ n, satisfies ψ(eMeM) = ψ(eN)ψ(eN) for all 2M,N ∈ n such that e2φ(M) = e2φ(N). Proof: IfM,N ∈ n such that e2φ(M) = e2φ(N), ψ(e2M) = e2φ(M) = e2φ(N) = eφ(N)eφ(N) = ψ(eN)ψ(eN). � Theorem 5.4: Let m, n ≥ 2 and φ : Mmn(C) → Mmn(C) be a linear map such that φ( mn) ⊆ mn. If tr(φ(A ⊕ B)) = tr(A ⊕ B) for all A ⊕ B ∈ mn, then the map ψ : GLmn(C) → GLmn(C) given by ψ(eM) := eφ(M), M ∈ mn, satisfies Det(ψ(A ⊗ B)) = Det(A ⊗ B) for all A ∈ GLm(C) and B ∈ GLn(C). LINEAR ANDMULTILINEAR ALGEBRA 2911 Proof: If A ∈ GLm(C), then there exists MA ∈ m such that A = eMA , and, similarly, if B ∈ GLn(C), there existsMB ∈ n such that B = eMB . We have A ⊗ B = exp(MA ⊕ MB). We note thatMA ⊕ 0n = MA ⊗ In ∈ mn and 0m ⊕ MB = Im ⊗ MB ∈ mn, so tr(φ(MA ⊕ MB)) = tr(φ(MA ⊕ 0n))+ tr(φ(0m ⊕ MB)) = tr(MA ⊕ 0n)+ tr(0m ⊕ MB) = tr(MA ⊕ MB). Letλ be an eigenvalue ofMA and μ be an eigenvalue ofMB. Then (λ+ μ)/2 is an eigenvalue of 1 2 (MA ⊕ MB), and all eigenvalues of 1 2 (MA ⊕ MB) have this form [9, Theorem 4.4.5]. Moreover, 12 (MA ⊕ MB) ∈ mn since Im(λ+ μ)/2 ∈ (−π ,π] whenever Im(λ), Im(μ) ∈ (−π ,π]. There exists a unique C ∈ mn such that A ⊗ B = eC. We have, A ⊗ B = eC = eMA⊕MB , det(A ⊗ B) = etr(C) = etr(MA⊕MB). Thus, det(eC/2)2 = det(e(MA⊕MB)/2)2. By Lemma 5.8, ψ(eC) = ψ(eC/2eC/2) = ψ(e 1 2 (A⊕B))ψ(e 1 2 (A⊕B)), and hence Det(ψ(A ⊗ B)) = Det(ψ(eC)) = Det(ψ(e 1 2 (MA⊕MB)))Det(ψ(e 1 2 (MA⊕MB))) = e 1 2 Tr(φ(MA⊕MB))e 1 2 Tr(φ(MA⊕MB)) · Rmn = eTr(MA⊕MB) · Rmn = Det(eMA⊕MB) = Det(A ⊗ B). � Theorem 5.2 and its corollaries provide the following corollaries. Corollary 5.5: Let m, n ≥ 2 and φ : Mmn(C) → Mmn(C) be a linear map such that φ( mn) ⊆ mn. If tr1(φ′(Imn)) = tr1(Imn) and tr2(φ′(Imn)) = tr2(Imn), then the map ψ : GLmn(C) → GLmn(C) given by ψ(eM) = eφ(M), M ∈ mn, satisfies Det(ψ(A ⊗ B)) = Det(A ⊗ B) for all A ∈ GLm(C) and B ∈ GLn(C). Lemma 5.9: Suppose that φ′(Imn) is a Kronecker sum. Then the following are equivalent: (1) tr1(φ′(Imn)) = tr1(Imn) and tr2(φ′(Imn)) = tr2(Imn), (2) φ′(Imn) = Imn, (3) tr(φ(E[mn] jk )) = δjk for all j, k ∈ {1, . . . ,mn}. 2912 A. FOŠNER ET AL. Proof: ((1)⇐⇒(2)) Suppose that φ′(Imn) = A ⊕ B is a Kronecker sum. If tr1(φ′(Imn)) = tr1(Imn), then tr(A)In + mB = mIn. (18) Similarly, if tr2(φ′(Imn)) = tr2(Imn) then tr(B)Im + nA = nIm. (19) Hence, A and B are both scalar matrices. Taking the trace of Equations (18) and (19) both yield that n tr(A)+ m tr(B) = mn, so that B = ( 1 − tr(A) m ) In, A = tr(A) m Im. It follows that φ′(Imn) = tr(A) m Im ⊗ In + Im ⊗ ( 1 − tr(A) m ) In = Im ⊗ In = Imn. The converse is immediate. ((2)⇐⇒(3)) Suppose that φ has the form φ(A) = mn∑ j,k,u,v=1 αjk;uvE [mn] jk AE[mn] uv (20) for some αjk;uv ∈ C. Then φ′(Imn) = mn∑ j,k,u,v=1 αuv;jkE [mn] jk E[mn] uv = mn∑ j,v=1 ( mn∑ k=1 αkv;jk ) E[mn] jv = Imn (21) if and only if ∑mn k=1 αkv;jk = δjv, where δjv is the Kronecker delta. On the other hand, tr(φ(E[mn] jk )) = tr ( mn∑ s,t,u,v=1 αst;uvE [mn] st E[mn] jk E[mn] uv ) = tr ( mn∑ s,v=1 αsj;kvE[mn] sv ) = mn∑ s=1 αsj;ks = δjk if and only if mn∑ k=1 αkv;jk = δjv. � LINEAR ANDMULTILINEAR ALGEBRA 2913 Corollary 5.6: Let m, n ≥ 2 and φ : Mmn(C) → Mmn(C) be a linear map such that φ( mn) ⊆ mn. If tr(φ(E[mn] jk )) = δjk for all j, k ∈ {1, . . . ,mn}, then the map ψ : GLmn(C) → GLmn(C) given by ψ(eM) = eφ(M), M ∈ mn, satisfies Det(ψ(A ⊗ B)) = Det(A ⊗ B) for all A ∈ GLm(C) and B ∈ GLn(C). Corollary 5.7: Let m, n ≥ 2 and φ : Mmn(C) → Mmn(C) be an RT-symmetric map such that φ( mn) ⊆ mn. If tr1(φ(Imn)) = tr1(Imn) and tr2(φ(Imn)) = tr2(Imn), then the map ψ : GLmn(C) → GLmn(C) given by ψ(eM) = eφ(M), M ∈ mn, satisfies Det(ψ(A ⊗ B)) = Det(A ⊗ B) for all A ∈ GLm(C) and B ∈ GLn(C). Corollary 5.8: Let m, n ≥ 2 and φ : Mmn(C) → Mmn(C) be a skew RT-symmetric map such that φ( mn) ⊆ mn. If tr1(φ(Imn)) = − tr1(Imn) and tr2(φ(Imn)) = − tr2(Imn), then the map ψ : GLmn(C) → GLmn(C) given by ψ(eM) = eφ(M), M ∈ mn, satisfies Det(ψ(A ⊗ B)) = Det(A ⊗ B) for all A ∈ GLm(C) and B ∈ GLn(C). 6. Conclusion We have considered three different notions of the determinant operation in Mmn(F), namely the block determinant, partial determinant and determinant-root. The block deter- minant and partial determinant-root retain the multiplicative property and exponential- determinant-trace relation, although only for Kronecker products in the case of the determinant-root. The results offer a method for obtaining non-linear preservers of determinant-roots of Kronecker products by utilizing the linear preservers of Kronecker sums. Acknowledgments Any opinions, findings and conclusions or recommendations expressed is that of the author(s), and the NRF accepts no liability whatsoever in this regard. Disclosure statement No potential conflict of interest was reported by the author(s). Funding Ajda Fošner was partially supported by the Slovenian Research Agency through the project No. J1- 2457. Yorick Hardy was supported by the National Research Foundation (NRF), South Africa. This work is partially based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers: 105968). 2914 A. FOŠNER ET AL. ORCID Y. 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Mich Math J. 1957;4(1):1–3. doi: 10.1307/mmj/1028990168 http://orcid.org/0000-0003-0264-8874 https://doi.org/10.1016/j.laa.2017.06.026 https://doi.org/10.13001/1081-3810.3688 https://doi.org/10.4153/CMB-1961-010-9 https://doi.org/10.1016/j.laa.2016.10.004 https://doi.org/10.2307/3620776 https://doi.org/10.1080/03081088108817379 https://doi.org/10.1016/S0377-0427(00)00393-9 https://doi.org/10.1080/03081087.2012.740029 https://doi.org/10.1016/j.jmaa.2016.08.037 https://doi.org/10.1016/j.laa.2019.09.003 https://doi.org/10.1017/S0004972700035711 https://doi.org/10.1307/mmj/1028990168 1. Introduction 2. Block determinants 2.1. Products 2.2. The exponential map 3. Partial determinants 3.1. Kronecker products 3.2. The exponential map 4. Determinant-roots 4.1. Fields with m-th roots of unity 4.2. The determinant-root on Dn(F) 4.3. Products 4.4. The exponential map 5. Non-linear preservers of determinants 5.1. RT-Symmetry 5.2. Linear trace preservers of Kronecker sums 5.3. Determinant preservers of Kronecker products 6. Conclusion Acknowledgments Disclosure statement Funding ORCID References << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /PageByPage /Binding /Left /CalGrayProfile () /CalRGBProfile (Adobe RGB \0501998\051) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.5 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /sRGB /DoThumbnails true /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 524288 /LockDistillerParams true /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments false /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings false /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Remove /UCRandBGInfo /Remove /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects true /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description << /ENU () >> >> setdistillerparams << /HWResolution [600 600] /PageSize [493.483 703.304] >> setpagedevice