Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

In this paper, we discuss the embedding problem for centrosymmetric matrices, which are higher order generalizations of the matrices occurring in strand symmetric models. These models capture the substitution symmetries arising from the double helix structure of the DNA. Deciding whether a transition matrix is embeddable or not enables us to know if the observed substitution probabilities are consistent with a homogeneous continuous time substitution model, such as the Kimura models, the Jukes-Cantor model or the general time-reversible model. On the other hand, the generalization to higher order matrices is motivated by the setting of synthetic biology, which works with different sizes of genetic alphabets.


Introduction
Phylogenetics is the study of evolutionary relationships among species that aims to infer the evolutionary history among them.In order to model evolution, we consider a phylogenetic tree, that is a directed acyclic graph depicting the evolutionary relationships amongst a selected set of taxa.Phylogenetic trees consist of vertices and edges.Vertices represent biological entities, while edges between vertices represent the evolutionary processes between the taxa.
In order to describe the real evolutionary process along an edge of a phylogenetic tree, one often assumes that the evolutionary data occurred following a Markov process.A Markov process is a random process in which the future is independent of the past, given the present.Under this Markov process, transitions between n states given by conditional probabilities are presented in a n ˆn Markov matrix M , namely a square matrix whose entries are nonnegative and rows sum to one.A well-known problem in probability theory is the so-called embedding problem which was initially posed by Elfving [18].The embedding problem asks whether given a Markov matrix M , one can find a real square matrix Q with rows summing to zero and non-negative off-diagonal entries, such that M " exppQq.The matrix Q is called a Markov generator.
In the complex setting, the embedding problem is completely solved by [23]; a complex matrix A is embeddable if and only if A is invertible.However, as our motivation arises from molecular models of evolution we are interested in the embedding problem over the real numbers, so from now on we will denote by M a real Markov matrix.It was shown by Kingman [31] that if an n ˆn real Markov matrix M is embeddable, then the matrix M has det M ą 0.Moreover, in the same work by Kingman it was shown that det M ą 0 is a necessary and sufficient condition for a 2 ˆ2 Markov matrix M to be embeddable.For 3 ˆ3 Markov matrices a complete solution of the embedding problem is provided in a series of papers [27,29,7,14], where the characterisation of embeddable matrices depends on the Jordan decomposition of the Markov matrix.For 4 ˆ4 Markov matrices the embedding problem is completely settled in a series of papers [8,10,36], where similarly to the 3 ˆ3 case the full characterisation of embeddable matrices is distinguished into cases depending on the Jordan form of the Markov matrices.
For the general case of n ˆn Markov matrices, there are several results; some presenting necessary conditions [18,31,38], while others sufficient conditions [27,19,21,17] for embeddability of Markov matrices.Moreover, the embedding problem has been solved for special n ˆn matrices with a biological interest such as equal-input and circulant matrices [3], group-based models [2] and time-reversible models [28].Despite the fact that there is no theoretical explicit solution for the embeddability of general n ˆn Markov matrices, there are results [10] that enable us to decide whether a n ˆn Markov matrix with distinct eigenvalues is embeddable or not.This is achieved by providing an algorithm that outputs all Markov generators of such a Markov matrix [10,35].
In this paper, we focus on the embedding problem for n ˆn matrices that are symmetric about their center and are called centrosymmetric matrices (see Definition 2.2).We also study a variation of the famous embedding problem called model embeddability, where apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure.For instance, for centrosymmetric matrices, model embeddability means that the rate matrix is also centrosymmetric.
The motivation for studying centrosymmetric matrices comes from evolutionary biology, as the most general nucleotide substitution model when considering both DNA strands admits any n ˆn centrosymmetric Markov matrix as a transition matrix, where n is the even number of nucleotides.For instance, by considering the four natural nucleotides A-T, C-G we arrive at the strand symmetric model, a well-known phylogenetic model whose substitution probabilities reflect the symmetry arising from the complementarity between the two strands that the DNA is composed of (see [12]).In particular, a strand symmetric model for DNA must have the following equalities of probabilities in the root distribution: and the following equalities of probabilities in the transition matrices pθ ij q θ AA " θ TT , θ AC " θ TG , θ AG " θ TC , θ AT " θ TA , θ CA " θ GT , θ CC " θ GG , θ CG " θ GC , θ CT " θ GA .
Therefore, the corresponding transition matrices of this model are 4 ˆ4 centrosymmetric matrices, usually called strand symmetric Markov matrices in this context.In the strand symmetric model there are less restrictions on the way genes mutate from ancestor to child compared to other widely known molecular models of evolution.In fact, special cases of the strand symmetric model are the group-based phylogenetic models such as the Jukes-Cantor (JC) model, the Kimura 2-parameter (K2P) and Kimura 3-parameter (K3P) models.The algebraic structure of strand symmetric models was initially studied in [12], where it was argued that strand symmetric models capture more biologically meaningful features of real DNA sequences than the commonly used group-based models, as for instance, in any group-based model, the stationary distribution of bases for a single species is always the uniform distribution, while computational evidence in [41] suggests that the stationary distribution of bases for a single species is rarely uniform, but must always satisfy the symmetries (1.1) arising from nucleotide complementarity, as assumed by the strand symmetric model.
In this article, we also explore higher order centrosymmetric matrices for which n ą 4, which is justified by the use of synthetic nucleotides.One of main goals of synthetic biology is to expand the genetic alphabet to include an unnatural or synthetic base pair.The more letters in a genetic system could possibly lead to an increased potential for retrievable information storage and bar-coding and combinatorial tagging [5].Naturally the four-letter genetic alphabet consists of just two pairs, A-T and G-C.In 2012, a genetic system comprising of three base pairs was introduced in [33].In addition to the natural base pairs, the third, unnatural or synthetic base pair 5SICS-MMO2 was proven to be functionally equivalent to a natural base pair.Moreover, when it is combined with the natural base pairs, 5SICS-MMO2 provides a fully functional six-letter genetic alphabet.Namely, six-letter genetic alphabets can be copied [45], polymerase chain reaction (PCR)-amplified and sequenced [40,44], transcribed to six-letter RNA and back to six-letter DNA [32], and used to encode proteins with added amino acids [4].This biological importance and relevance of the above six-letter genetic alphabets motivates us to particularly study the 6 ˆ6 Markov matrices describing the probabilities of changing base pairs in the six-letter genetic system in Section 6.When considering both DNA strands, each substitution is observed twice due to the complementarity between both strands, and hence the resulting transition matrix is centrosymmetric.
Moreover there are other synthetic analogs to natural DNA which justify studying centrosymmetric matrices for n ą 6.For instance, hachimoji DNA is a synthetic DNA that uses four synthetic nucleotides B, Z, P, S in addition to the four natural ones A,C, G, T. With the additional four synthetic ones, hachimoji DNA forms four types of base pairs, two of which are unnatural: P binds with Z and B binds with S. The complementarity between both strands of the DNA implies that the transition matrix is centrosymmetric.Moreover, the research group responsible for the hachimoji DNA system had also studied a synthetic DNA analog system that used twelve different nucleotides, including the four found in DNA (see [43]).Although the biological models which motivate the study of centrosymmetric matrices in this paper require n to be an even number due to the double-helix structure of DNA, in Section 5, we include the case of n being odd for completeness.
Apart from embeddability, namely existence of Markov generators, it is also natural to ask about uniqueness of a Markov generator which is called the rate identifiability problem.Identifiability is a property which a model must satisfy in order for precise statistical inference to be possible.A class of phylogenetic models is identifiable if any two models in the class produce different data distributions.In this article, we further develop the results on rate identifiability of the Kimura two parameter model [8] to study rate identifiability for strand symmetric models.We also show that there are embeddable strand symmetric Markov matrices with non identifiable rates, namely the Markov generator is not unique.Moreover, we show that strand symmetric Markov matrices are not generically identifiable, that is, there exists a positive measure subset of strand symmetric Markov matrices containing embeddable matrices whose rates are not identifiable.
This paper is organised as following.In Section 2, we introduce the basic definitions and results on embeddability.In Section 3, we give a characterisation for a 4 ˆ4 centrosymmetric Markov matrix M (also known as a strand symmetric Markov matrix) with four distinct real nonnegative eigenvalues to be embeddable providing necessary and sufficient conditions in Theorem 3.7, while we also discuss their rate identifiability property in Proposition 3.9.Moreover in Section 4, using the conditions of our main result Theorem 3.7, we compute the relative volume of all strand symmetric Markov matrices relative to the strand symmetric Markov matrices with positive eigenvalues and ∆ ą 0, as well as the relative volume of all strand symmetric Markov matrices relative to the strand symmetric Markov matrices with four distinct eigenvalues and ∆ ą 0. We also compare the results on relative volumes obtained using our method with the algorithm suggested in [10] to showcase the advantages of our method.In Section 5, we study higher order centrosymmetric matrices and motivate their use in Section 6 by exploring the case of synthetic nucleotides where the phylogenetic models admit 6 ˆ6 centrosymmetric mutation matrices.Finally, Section 7 discusses implications and possibilities for future work.

Preliminaries
In this section we will introduce the definitions and results that will be required throughout the paper.We will denote by M n pKq the set of nˆn square matrices with entries in the field K " R or C. The subset of non-singular matrices in M n pKq will be denoted by GL n pKq.Definition 2.1.We will call Markov (or transition) matrices the non-negative real square matrices with rows summing to one.Rate matrices are real square matrices with rows summing to zero and non-negative off-diagonal entries.
In this paper, we are focusing on a subset of Markov matrices called centrosymmetric Markov matrices.
Definition 2.2.A real n ˆn matrix A " pa i,j q is said to be centrosymmetric (CS) if for every 1 ď i, j ď n.Definition 2.2 reveals that a CS matrix is nothing more than a square matrix which is symmetric about its center.This class of matrices has been previously studied, for instance, in [1, page 124] and [42].Examples of CS matrices for n " 5 and n " 6, are the following two matrices respectively: .
The class of CS matrices plays an important role in the study of Markov processes since they are indeed transition matrices for some processes in evolutionary biology.For instance, in [30], centrosymmetric matrices are used to study the random assortment phenomena of subunits in chromosome division.Furthermore, in [39], the same centrosymmetric matrices appear as the transition matrices in the model of subnuclear segregation in the macronucleus of ciliates.Finally, the work [26] examines a special case of random genetic drift phenomenon, which consists of a population consisting of individuals that are able to produce a single type of gamete and the transition matrices of the associated Markov chain are given by centrosymmetric matrices.
The embedding problem is directly related to the notions of matrix exponential and logarithm which we introduce for completeness below.Definition 2.3.We define the exponential exppAq of a matrix A, using the Taylor power series of the function f pxq " e x , as exppAq " where A 0 " I n and I n denotes the n ˆn identity matrix.If A " P diagpλ 1 , . . ., λ n q P ´1 is an eigendecomposition of A, then exppAq " P diagpe λ 1 , . . ., e λn q P ´1.Given a matrix A P M n pKq, a matrix B P M n pKq is said to be a logarithm of A if exppBq " A. If v is an eigenvector corresponding to the eigenvalue λ of A, then v is an eigenvector corresponding to the eigenvalue e λ of exppAq.
A Markov matrix M is called embeddable if it can be written as the exponential of a rate matrix Q, namely M " exppQq.Then any rate matrix Q satisfying the equation M " exppQq is called a Markov generator of M .Remark 2.4.We should note here that embeddable Markov matrices occur when we assume a continuous time Markov chain, in which case the Markov matrices have the form M " expptQq, where t ě 0 represents time and Q is a rate matrix.However, in the rest of the paper, we assume that t is incorporated in the rate matrix Q.
The existence of multiple logarithms is a direct consequence of the distinct branches of the logarithmic function in the complex field.Definition 2.5.Given z P CzR ď0 and k P Z, the k-th branch of the logarithm of z is log k pzq :" log |z|`pArgpzq`2πkqi, where log is the logarithmic function on the real field and Argpzq P p´π, πq denotes the principal argument of z.The logarithmic function arising from the branch log 0 pzq is called the principal logarithm of z and is denoted as logpzq.
It is known that if A is a matrix with no negative eigenvalues, then there is a unique logarithm of A all of whose eigenvalues are given by the principal logarithm of the eigenvalues of A [23,Theorem 1.31].We refer to this unique logarithm as the principal logarithm of A, denoted by LogpAq.
By definition, the Markov generators of a Markov matrix M are those logarithms of M that are rate matrices.In particular they are real logarithms of M .The following result enumerates all the real logarithms with rows summing to zero of any given Markov matrix with positive determinant and distinct eigenvalues.Therefore, all Markov generators of such a matrix are necessarily of this form.
In this paper, we focus on the embedding problem for the class of centrosymmetric matrices.In Section 3, we will first study the embeddability of 4 ˆ4 centrosymmetric Markov matrices, which include the K3P, K2P and JC Markov matrices.In Section 5 and Section 6, we will further study the embeddability of higher order centrosymmetric Markov matrices.

Embeddability of 4 ˆ4 centrosymmetric matrices
In this section, we begin our study by analyzing the embeddability of 4ˆ4 centrosymmetric matrices also known as strand symmetric matrices.We will provide necessary and sufficient conditions for 4 ˆ4 CS matrices to be embeddable.Moreover, we will discuss their rate identifiability problem as well.
Phylogenetic evolutionary models whose mutation matrices are 4 ˆ4 centrosymmetric matrices are also called strand symmetric Markov models.The transition matrices in the strand symmetric model are assumed to have the form In the case of the K2P matrices, we additionally have m 12 " m 13 , while in the case of JC matrices, m 12 " m 13 " m 14 .It can be easily seen that K3P, K2P, and JC Markov (rate) matrices are centrosymmetric.
Let us define the following matrix compare [11,Section 6].For a 4 ˆ4 CS Markov matrix M , we define F pM q :" S ´1M S. By direct computation, it can be checked that F pM q is a block diagonal matrix where We will then define two matrices, M 1 :" ˙, which are the upper and lower block matrices in (3.2), respectively.Similarly, the rate matrices in strand symmetric models are assumed to have the 4 ˆ4 centrosymmetric form Q " ¨q11 q 12 q 13 q 14 q 21 q 22 q 23 q 24 q 24 q 23 q 22 q 21 q 14 q 13 q 12 q 11 ‹ ‹ ‚ , where q 11 `q12 `q13 `q14 " 0 " q 21 `q22 `q23 `q24 and q ij ě 0 for i ‰ j.
We will then define two matrices, ˙, which are the upper and lower block matrices in (3.4), respectively.
The following results provide necessary conditions for a 4 ˆ4 CS Markov matrix to be embeddable.
Proposition 3.3.Given two matrices A " pa ij q, B " pb ij q P M 2 pRq, consider the block-diagonal matrix C " diagpA, Bq.Then the following statements hold: Then ii) follows from the above expression of F ´1pQq and the fact that rows of Markov matrices add to 1 and the entries are non-negative, while iii) similarly follows from the fact that the rows of rate matrices add to zero and the off-diagonal entries are non-negative.
For any 4ˆ4 CS Markov matrix M " pm ij q, let us recall that by (3.2), M is block-diagonalizable via the matrix S. In the rest of this section, we will study both the upper and the lower block matrices of F pM q more closely.Studying the upper and lower blocks allows us to establish the main result of the embeddability criteria for 4 ˆ4 CS Markov matrices.This block-diagonalization reduces our analysis to studying the logarithms of both the upper and the lower block matrices which have size 2 ˆ2.This result will be presented in Theorem 3.7.

Upper block
As we have seen in (3.2), the upper block of F pM q is given by the 2ˆ2 matrix M 1 " ´1˙, then Hence, by Proposition 2.6, any logarithm of M 1 can be written as for some integers k 1 and k 2 .Let p " logpλ `µ ´1q, q " 1 ´λ, and r " 1 ´µ.Then Proof.For fixed k 1 and k 2 , the eigenvalues of is a real matrix if and only if λ 1 , λ 2 P R. Finally, λ 1 P R if and only if k 1 " 0 and λ 2 P R if and only if k 2 " 0 and λ `µ ą 1.

Lower block
The lower block of F pM q is given by the matrix generally not a Markov matrix.The discriminant of the characteristic polynomial of M 2 is given by ∆ :" pα ´βq 2 `4α with α, β, α 1 , β 1 defined as in (3.3).If ∆ ą 0, then M 2 has two distinct real eigenvalues and if ∆ ă 0, then M 2 has a pair of conjugated complex eigenvalues.Moreover, if ∆ " 0, then M 2 has either 2 ˆ2 Jordan block or a repeated real eigenvalue.We will assume that ∆ ‰ 0 so that M 2 diagonalizes into two distinct eigenvalues.
1.If ∆ ą 0, then Impl 3 q " 2k 3 π and Impl 4 q " 2k 4 π.Moreover, Repl 3 q ‰ Repl 4 q.Since l 3 and l 4 are the eigenvalues of L M 2 k 3 ,k 4 , this implies that l 3 ‰ l 4 .In particular, L M 2 k 3 ,k 4 is a real matrix if and only if both l 3 and l 4 are real.
2. Let us assume ∆ ă 0 and take z " pα`βq`?∆ 2 .Fixing k 3 , k 4 P Z, the eigenvalues of L M 2 k 3 ,k 4 are l 3 " logpzq `2k 3 πi and l 4 " logpzq `2k 4 πi " Logpzq `2k 4 πi, which are both complex numbers.Thus, L M 2 k 3 ,k 4 is real if and only if l 3 " l 4 .Hence, k 4 " ´k3 .Conversely, k 4 " ´k3 implies that l 3 `l4 " 2Repl 3 q P R and l 3 ´l4 ?We note that the subset of 4 ˆ4 CS Markov matrix with repeated eigenvalues (diagonalizing matrix with repeated eigenvalues or a Jordan block of size greater than 1) have zero measure.Therefore generic 4 ˆ4 Markov matrices have no repeated eigenvalues, and hence we are going to assume the eigenvalues to be distinct .In particular, we are assuming that M diagonalizes.Furthermore, since we want M to have real logarithms and have no repeated eigenvalues, we need the real eigenvalues to be positive.
The following theorem characterizes the embeddability of a 4ˆ4 CS Markov matrix with positive and distinct eigenvalues.Furthermore, the theorem guarantees that a 4 ˆ4 CS Markov matrix is embeddable if and only if it admits a CS Markov generator.In particular, the characterization of the embeddability of a CS matrix is equivalent when restricting to rate matrices satisfying the symmetries imposed by the model (model embeddability) than when restricting to all possible rate matrices (embedding problem).
In particular, any real logarithm of M is also a 4 ˆ4 CS matrix whose entries q 11 , . . ., q 24 are given by: Proof.Let us note that M " S ¨diagpP 1 , P 2 q ¨diagp1, λ 1 , λ 2 , λ 3 q ¨diagpP ´1 1 , P ´1 2 q ¨S´1 .Since we assume that the eigenvalues of M are distinct, according to Proposition 2.6, any logarithm of M can be written as The last equation and the fact that S and S ´1 are real matrices imply that Q will be real if and only if both L M 1 k 1 ,k 2 and L M 2 k 3 ,k 4 are real.Here L M 1 k 1 ,k 2 is the upper block given in (3.7) and L M 2 k 3 ,k 4 is the lower block defined in (3.9).By Lemma 3.4, L M 1 k 1 ,k 2 being a real logarithm implies that k 1 " k 2 " 0 and λ `µ ą 1.Then L M 2 k 3 ,k 4 being a real matrix, according to Lemma 3.5, implies that k 3 " k 4 " 0 if ∆ ą 0, while k 4 " ´k3 if ∆ ă 0. Therefore, the upper block is L M 1 0,0 and the lower block will be L M 2 k,´k , for k " k 3 completing the proof.Now we are interested in knowing when the real logarithm of a 4 ˆ4 CS Markov matrix is a rate matrix.Using the same notation as in Theorem 3.6 we get the following result.
Theorem 3.7.A diagonalizable 4ˆ4 CS Markov matrix M with distinct eigenvalues is embeddable if and only if the following conditions hold for k " 0 if ∆ ą 0 or for some k P Z if ∆ ă 0: Proof.The logarithm of a 4 ˆ4 CS Markov matrix will depend on whether ∆ ą 0 or ∆ ă 0. In particular, it will depend on whether the eigenvalues λ 2 and λ 3 are real and positive or whether they are conjugated complex numbers.
In particular, Theorem 3.6 together with Proposition 3.3 imply that a real logarithm of M is a rate matrix if and only if Furthermore, the conditions λ 1 ą 0 comes from Lemma 3.4.The proof is now complete.
Remark 3.8.According to Theorem 3.7 the embeddability of a 4 ˆ4 CS Markov matrix M with distinct positive eigenvalues can be decided by checking six inequalities depending on the entries of M .However, if M has non-real eigenvalues then one has to check infinitely many groups of inequalities, one for each value of k P Z.It is enough that one of those systems is consistent to guarantee that M is embeddable.Theorem 5.5 in [10] provides boundaries for the values of k for which the corresponding inequalities may hold.
Let us take a look at the class of K3P matrices which is a special case of strand symmetric matrices.Indeed, for a K3P matrix M " pm ij q, we have that Suppose that a K3P-Markov matrix M " pm ij q is K3P-embeddable, i.e.M " exppQq for some K3P-rate matrix Q. Recall that the eigenvalues of M are 1, p :" m 11 `m12 ´m13 ´m14 , q :" m 11 ´m12 `m13 ´m14 and r :" m 11 ´m12 ´m13 `m14 .
In particular, we see that ∆ ą 0 unless m 12 " m 13 .Moreover, x " log r, y " log q, z " log p, α 1 " The inequalities in Theorem 3.7 can be spelled out as follows: In the last part of this section, we discuss the rate identifiability problem for 4 ˆ4 centrosymmetric matrices.If a centrosymmetric Markov matrix arises from a continuous-time model, then we want to determine its corresponding substitution rates.Namely, given an embeddable 4 ˆ4 CS matrix, we want to know if we can uniquely identify its Markov generator.
It is worth noting that Markov matrices with repeated real eigenvalues may admit more than one Markov generator (e.g.examples 4.2 and 4.3 in [8] show embeddable K2P matrices with more than one Markov generator).Nonetheless, this is not possible if the Markov matrix has distinct eigenvalues, because in this case its only possible real logarithm would be the principal logarithm [15].As one considers less restrictions in a model, the measure of the set of matrices with repeated real eigenvalues decreases, eventually becoming a measure zero set.For example, this is the case within the K3P model, where both its submodels (the K2P model and the JC model) consist of matrices with repeated eigenvalues and have positive measure subsets of embeddable matrices with non-identifiable rates.However, when considering the whole set of K3P Markov matrices, the subset of embeddable matrices with more than one Markov generator has measure zero (see Chapter 4 in [35]).Nevertheless, this behaviour only holds if the Markov matrices within the model have real eigenvalues.Proposition 3.9.There is a positive measure subset of 4 ˆ4 CS Markov matrices that are embeddable and whose rates are not identifiable.Moreover, all the Markov generators of the matrices in this set are also CS matrices.
A straightforward computation shows that M is a CS Markov matrix and Q is a CS rate matrix.Moreover they both have non-zero entries.By applying the exponential series to Q, we get that exppQq " M .That is M is embeddable and Q is a Markov generator of M .Since Q is a rate matrix, so is Qt for any t P R ě0 .Therefore, exppQtq is an embeddable Markov matrix, because the exponential of any rate matrix is necessarily a Markov matrix.See [34,Theorem 4.19] for more details.Moreover, we have that so S ´1 exppQtqS is a 2-block diagonal matrix.Hence, by Proposition 3.3 we have that exppQtq is an embeddable strand symmetric Markov matrix for all t P R ą0 .Now, let us define V " P diagp0, 0, 2πi, ´2πiq P ´1.Note that Q and V diagonalize simultaneously via P and hence they commute.Therefore, exppQ `V q " exppQq exppV q " M I 4 " M by the Baker-Campbell-Haussdorff formula.Moreover, exppQt `kV q " exppQtq exppkV q " exppQtqI 4 " exppQtq for all k P Z.Note that kV is a bounded matrix for any given k and hence, given t large enough, it holds that Qt `mV is a rate matrix for any m between 0 and k.
This shows that, for t large enough, exppQtq is an embeddable CS Markov matrix with at least k `1 different CS Markov generators.Moreover, exppQtq and all its generators have no null entries by construction and they can therefore be perturbed as in Theorem 3.3 in [9] to obtain a positive measure subset of embeddable CS Markov matrices that have k `1 CS Markov generators.Such perturbation consists of small enough changes on the real and complex parts of the eigenvalues and eigenvectors of M (other than the eigenvector p1, . . ., 1q and its corresponding eigenvalue 1.) Remark 3.10.Using the same notation as in the proposition above and given C P GL 2 pCq, let us define Since QpI 2 q " Q is a CS rate matrix with no null entries, so is QpCq for C P GL 2 pCq close enough to I 2 .Moreover, by construction we have that expp2tQpCqq " expp2tQq for all t P N.

Volumes of ˆCS Markov matrices
In this section, we compute the relative volumes of embeddable 4 ˆ4 CS Markov matrices within some meaningful subsets of Markov matrices.The aim of this section is to describe how large the different sets of matrices are compared to each other.Let V M arkov 4 be the set of all 4 ˆ4 CS Markov matrices.We use the following description .Let V `be the set of all CS Markov matrices having real positive eigenvalues, where ∆ " pp1 ´e ´2g ´hq ´p1 ´b ´c ´2dqq 2 `4pe ´hqpb ´cq, is the discriminant of the matrix M 2 as stated in Section 3. We have
Finally, we consider the following two biologically relevant subsets of V M arkov

4
. Let V DLC be the set of diagonally largest in column (DLC) Markov matrices, which is the subset of V M arkov 4 containing all CS Markov matrices such that the diagonal element is the largest element in each column.These matrices are related to matrix parameter identifiability in phylogenetics [13].Secondly, we let V DD be the set of diagonally dominant (DD) Markov matrices, which is the subset of V M arkov 4 matrices containing all CS Markov matrices such that in each row the diagonal element is at least the sum of all the other elements in the row.Biologically, the subspace V DD consists of matrices with probability of not mutating at least as large as the probability of mutating.If a diagonally dominant matrix is embeddable, it has an identifiable rate matrix [16,27].By the definition of each set, we have the inclusion V DD Ď V DLC .
Remark 4.1.The sets V `, V em`, V DLC , V DD that we consider in this section are all subsets of the set V M arkov 4 of all 4 ˆ4 CS Markov matrices, but we can use the same definition to refer to the equivalent subsets of n ˆn CS Markov matrices.Therefore, we will use the same notation V `, V em`, V DLC , V DD to refer to the equivalent subsets of the set V M arkov n of n ˆn CS Markov matrices without confusion in the following sections.
In the rest of this section, the number vpAq denotes the Euclidean volume of the set A. By definition, V M arkov 4 , V DLC and V DD are polytopes, since they are defined by the linear inequalities in R 6 .Hence, we can use Polymake [20] to compute their exact volumes and obtain that vpV M arkov 4 q " 1 36 , vpV DLC q " 1 576 and vpV DD q " 1 2304 .
Hence, we see that V DLC and V DD constitute roughly only 6.25% and 1.56% of V M arkov

4
, respectively.On the other hand, we will estimate the volume of the sets V `, V em`, V DLC X V `, V DLC X V em`, V DD X V `, and V DD X V em`u sing the hit-and-miss Monte Carlo integration method [22] with sufficiently many sample points in Mathematica [25].Theoretically, Theorem 3.7 enables us to compute the exact volume of these relevant sets.For example in the case of K3P matrices, such exact computation of volumes has been feasible in [37].However, while for the K3P matrices, the embeddability criterion is given by three quadratic polynomial inequalities, in the case of CS matrices the presence of nonlinear and nonpolynomial constraints imposed on each set, makes the exact computation of the volume of these sets intractable.Therefore, we need to approximate the volume of these sets.Given a subset A Ď V M arkov 4 , the volume estimate of vpAq computed using the hit-and-miss Monte Carlo integration method with n sample points is given by the number of points belonging to A out of n sample points.For computational purposes, in the formula of φp0q and εp0q, we use the fact that y ´z " log ˜p2 ´b ´c ´2d ´e ´2g ´hq `?∆ p2 ´b ´c ´2d ´e ´2g ´hq ´?∆ ¸.
The results of these estimations using the hit-and-miss Monte Carlo integration implemented in Mathematica with n sample points are presented in Table 1, while Table 2 provides an estimated volume ratio between relevant subsets of centrosymmetric Markov matrices using again the hit-and-miss Monte Carlo integration with n sample points.In Table 1, we firstly generate n centrosymmetric matrices whose off-diagonal entries were sampled uniformly in r0, 1s and forced the rows of the matrix to sum to one.Out of these n matrices, we test how many of them are actually Markov matrices (i.e. the diagonal entries are non-negative) and then out of these how many have positive eigenvalues..In particular, for n "10 7 sample points containing 277628 centrosymmetric Markov matrices, Table 2 suggests that there are approximately 1.7% of centrosymmetric Markov matrices with distinct positive eigenvalues that are embeddable.Moreover, we can see that for n " 10 7 , out of all embeddable centrosymmetric Markov matrices with distinct positive eigenvalues, almost all are diagonally largest in column, while only 28% are diagonally dominant.vpV em`q vpV `q 0.130435 0.177083 0.17959 0.170429 vpV DLC XV em`q vpV DLC XV `q 0.157895 0.220779 0.231668 0.221289 vpV DLC XV em`q vpV `q 0.130435 0.177083 0.178589 0.17004 vpV DLC XV em`q vpV em`q 1 1 0.994429 0.997721 vpV DD XV em`q vpV DD XV `q 0.333333 0.483871 0.400763 0.349948 vpV DD XV em`q vpV `q 0.0434783 0.078125 0.052563 0.0490753 vpV DD XV em`q vpV em`q 0.333333 0.441176 0.292479 0.287952 An alternative approach for approximating the number of embeddable matrices within the model is to use Algorithm 5.8 in [10] to test the embeddability of the sample points.Tables 4 and  5 below are analogous to Tables 1 and 2, but Table 4 was obtained using the sampling method in [35, Appendix A], while using either Algorithm 5.8 in [10] or the inequalities in Theorem 3.7 yields identical results which are provided in Table 4 and Table 5.
We used the python implementation of Algorithm 5.8 in [10] provided in [35, Appendix A] and modified it to sample on the set of 4 ˆ4 CS Markov matrices with positive eigenvalues.The original sampling method used in [35, Appendix A] consisted of sampling uniformly on the set of 4 ˆ4 centrosymmetric-Markov matrices and what we did is keep sampling until we got n samples (or as many samples as we require) with positive eigenvalues.
Despite the fact that Theorem 3.7 and Algorithm 5.8 in [10] were originally implemented using different programming languages (Wolfram Mathematica and Python respectively) and were tested with different sample sets, the results obtained are quite similar as illustrated by Tables 2 and 5.In fact, when we are applying both Algorithm 5.8 in [10] and Theorem 3.7 on the same sample set in Table 3, we are obtaining identical results which are displayed in Tables 4 and 5.
Table 3: Number of samples in V `, V DLC X V `, and V DD X V `obtained by using the sampling method in [35,Appendix A].
Samples in V `10 4 10 5 10 6 10 7 Samples in V DLC X V `8531 85446 854709 8549100 Samples in V DD X V `1464 14538 144546 1448720 Table 4: Number of samples in V em`, V DLC X V em`a nd V DD X V em`o btained by applying either Theorem 3.7 or the results in [10] on the sample set in Table 3.
Samples in V em`1 877 18663 185357 1862413 Samples in V DLC X V em`1 869 18586 184555 1854592 Samples in V DD X V em`5 16 5164 50058 504304 Table 5: Relative volumes ratio between the relevant subsets obtained using hit-and-miss method and either Algorithm 5.8 in [10] or Theorem 3.7.The volumes were estimated as the quotient of the sample sizes in Tables 3 and 4. n 10 4 10 5 10 6 10 7 vpV em`q vpV `q 0.1877 0.18663 0.185357 0.1862413 vpV DLC XV em`q vpV DLC XV `q 0.2191 0.2175 0.2159 0.2169 vpV DLC XV em`q vpV `q 0.1869 0.18586 0.184555 0.1854592 vpV DLC XV em`q vpV em`q 0.9957 0.9959 0.99567 0.99580 vpV DD XV em`q vpV DD XV `q 0.3524 0.3552 0.3463 0.3481 vpV DD XV em`q vpV `q 0.0516 0.05164 0.050058 0.0504 vpV DD XV em`q vpV em`q 0.2749 0.2767 0.2701 0.2708 It is worth noting that the embeddability criteria given in Theorem 3.7 use inequalities depending on the entries of the matrix, whereas Algorithm 5.8 in [10] relies on the computation of its principal logarithm and its eigenvalues and eigenvector, which may cause numerical issues when working with matrices with determinant close to 0. What is more the computation of logarithms can be computationally expensive.As a consequence, the algorithm implementing the criterion for embeddability arising from Theorem 3.7 is faster.Table 6 shows the running times for the implementation of both embeddability criteria used to obtain Table 5.
Table 6: Running times for the Python implementation of the embeddability criterion arising from Theorem 3.7 and from Algorithm 5.8 in [10].The simulations were run using a computer with 8GB of memory.The Python implementation of Algorithm 5.8 in [10] provided in [35, Appendix A] can also be used to test the embeddability of any 4 ˆ4 CS Markov matrix (including those with non-real eigenvalues) without modifying the embeddability criteria.All it takes is a suitable sample set.As hinted in Remark 3.8, this would also be possible using the embedability criterion in Theorem 3.7 together with the boundaries for k provided in [10,Theorem 5.5].Table 7 shows the results obtained when applying Algorithm 5.8 in [10] to a set of 10 As most DLC and DD matrices have positive eigenvalues, the proportion of embedabbile matrices within these subsets is almost the same when admitting matrices with non-positive eigenvalues (as in Table 7 instead of only considering matrices with positive eigenvalues as we did in Tables 2  and 5. On the other hand, the proportion of 4 ˆ4 embeddable CS matrices is much smaller in this case.

Centrosymmetric matrices and generalized Fourier transformation
In Section 3 and 4 we have seen the embeddability criteria for 4 ˆ4 centrosymmetric Markov matrices and the volume of their relevant subsets.In this section, we are extending this framework to larger matrices.The importance of this extension is relevant to the goal of synthetic biology which aims to expand the genetic alphabet.For several decades, scientists have been cultivating ways to create novel forms of life with basic biochemical components and properties far removed from anything found in nature.In particular, they are working to expand the number of amino acids which is only possible if they are able to expand the genetic alphabet (see for example [24]).

Properties of centrosymmetric matrices
For a fixed n P N, let V n denote the set of all centrosymmetric matrices of order n.Moreover, let V M arkov n and V rate n denote the set of all centrosymmetric Markov and rate matrices of order n, respectively.As a subspace of the set of all n ˆn real matrices, for n even, dimpV n q " n 2 2 while for n odd, dimpV n q " t n 2 upn `1q `1.We will now mention some geometric properties of the sets V M arkov n and V rate n .Furthermore, for any real number x, txu and rxs denote the floor and the ceiling function of x, respectively.Proposition 5.1.
1.For n even, is a Cartesian product of n 2 standard pn´1q-simplices and its volume is ě0 is a Cartesian product of t n 2 u standard pn ´1q-simplices and the t n 2 u-simplex with vertices t0, e i 2 u 1ďiďt n 2 u Y te t n 2 u`1 u, where e i is the i-th standard unit vector in R n , which is the vector that has 1 as the i-th component and zeros elsewhere.Hence, the volume of V M arkov n is Proof.Here we consider the following identification for an n ˆn centrosymmetric matrix M .For n even, M can be thought as a point pM 1 , . . ., M n 2 q P pR n ě0 q n 2 where the point M i P R n ě0 corresponds to the i-th row of M .Similarly, for n odd, we identify M as a point in pR n ě0 q t n 2 u ˆRt n 2 u`1 ě0 . Since M is a Markov matrix, under this identification, each point M i lies in some simplices.Therefore, V M arkov n is a Cartesian product of some simplices.For n even, these simplices are the standard pn ´1q-dimensional simplex: For n odd and 1 ď i ď t n 2 u, the point M i belongs to standard pn ´1q-simplex above and the point M t n 2 u`1 belongs to the simplex (5.2) We now compute the volume of V M arkov n .Let us recall the fact that the volume of the Cartesian product of spaces is equal to the product of volumes of each factor space if the volume of each factor space is bounded.Moreover, the pn ´1q-dimensional volume of the standard simplex in Equation (5.1) in R n´1 is 1 pn´1q! .For n even, the statement follows immediately.For n odd, we use the fact that the t n 2 u-dimensional volume of the simplex in Equation (5.2) is For the second statement, we use the fact that if Q is a rate matrix, then q ii " ´řj‰i q ij where q ij ě 0 for i ‰ j.
In the rest of this section, let J n be the n ˆn anti-diagonal matrix, i.e. the pi, jq-entries are one if i `j " n `1 and zero otherwise.The following proposition provides some properties of the matrix J n that can be checked easily.Proposition 5.2.Let A " pa ij q P M n pRq.Then 1. pAJ n q ij " a i,n`1´j and pJ n Aq ij " a n`1´i,j .

2.
A is a centrosymmetric matrix if only if J n AJ n " A.
In Section 3, we have seen that 4 ˆ4 CS matrices can be block-diagonalized through the matrix S. Now we will present a construction of generalized Fourier matrices to block-diagonalize any centrosymmetric matrices.Let us consider the following recursive construction of the n ˆn matrix S n : , for n ě 3.
(5.3) Proposition 5.3.For each natural number n ě 3, S n is invertible and its inverse is given by Proof.The proposition easily follows from the definition of S n .Namely, The following proposition provides another block decomposition of the matrix S n and its inverse.

Using these block partitions, S ´1
n " 1 2 S n for n even, while S ´1 n " ‚ for n odd.
Proof.The proof follows from induction on n and the fact that J 2 n " I n .
We will call a vector v P R n symmetric if v i " v n`1´i for every 1 ď i ď n, i.e.J n v " v.Moreover, we call a vector w P R n anti-symmetric if v i " ´vn`1´i for every 1 ď i ď n, i.e.J n v " ´v.The following technical proposition will be used in what follows in order to simplify a centrosymmetric matrix.Proposition 5.5.Let n ě 2. Let v P R n be a symmetric vector and w P R n be an anti-symmetric vector.3. Then the sum of the entries of S n v and v T S n is the sum of the entries of v.

Then the sum of the entries of S ´1
n v and v T S ´1 n is the sum of the first r n 2 s entries of v. Proof.We will only prove the first part of item (1) in the proposition using mathematical induction on n.The base case for n " 2 can be easily obtained.Suppose now that the proposition holds for all k ă n.Let v " ¨v1 v 1 v 1 ‚ P R n be a symmetric element.Then v 1 P R n´2 is also symmetric.By direct computation we obtain The last t n´2 2 u entries of S n´2 v 1 are zero.Thus, the last t n´2 2 u `1 " t n 2 u entries of S n v are zero as well.The proof of the other statements can be obtained analogously using induction.In particular, let us note that the proof given for item (1) directly implies item (3).

Logarithms of centrosymmetric matrices
For the special structure encoded by the centrosymmetric matrices, one may ask whether they have logarithms which are also centrosymmetric.In this section, we provide some answers to this question.
Theorem 5.8.Let A P M n pRq be a CS matrix.Then A has a CS logarithm if and only if both the upper block matrix A 1 and the lower block matrix A 2 in Lemma 5.6 admit a logarithm.
Proof.Suppose that A has a centrosymmetric logarithm Q.By Lemma 5.6, F n pAq " diagpA 1 , A 2 q and F n pQq " diagpQ 1 , Q 2 q.Then exppQq " A implies that exppQ 1 q " A 1 and exppQ 2 q " A 2 .Hence, A 1 and A 2 admit a logarithm.Conversely, suppose that A 1 and A 2 admit a logarithm Q 1 and Q 2 , respectively.Then the matrix diagpQ 1 , Q 2 q is a logarithm of the matrix diagpA 1 , A 2 q.By Lemma 5.7, the matrix F ´1 n pdiagpQ 1 , Q 2 qq is a centrosymmetric logarithm of A. Proof.Let us suppose that LogpAq is not centrosymmetric matrix.Define the matrix Q " J n pLogpAqqJ n .Then Q ‰ LogpAq since LogpAq is not centrosymmetric.It is also clear that exppQq " A. Moreover, since J 2 n " I n , the matrices LogpAq and Q have the same eigenvalues.Therefore, Q is also a principal logarithm of A, a contradiction to the uniqueness of principal logarithm.Hence, LogpAq must be centrosymmetric.
The following theorem characterizes the logarithms of any invertible CS Markov matrices.
Theorem 5.11.Let A P M n pRq be an invertible CS Markov matrix.Let A 1 " N 1 D 1 N ´1 1 where D 1 " diagpR 1 , R 2 , . . ., R l q is a Jordan form of A 1 , where A 1 is the upper block matrix in Lemma 5.6.Similarly, let A 2 " N 2 D 2 N ´1 2 where D 2 " diagpT 1 , T 2 , . . ., T l q is a Jordan form of A 2 where A 1 is the lower block matrix in Lemma 5.6.Then A has a countable infinitely many logarithms given by where N :" diagpN 1 , N 2 q and D :" diagpD 1 1 , D 1 2 q, and D 1 i denotes a logarithm of D i .In particular, these logarithms of A are primary functions of A.
For the definition of primary function of a matrix, we refer the reader to [23].The above theorem says that the logarithms of a nonsingular centrosymmetric matrix contains a countable infinitely many primary logarithms and they are centrosymmetric matrices as well.
Finally, we will present a necessary condition for embeddability of CS Markov matrices in higher dimensions.
Proof.The matrix A is embeddable if and only if it admits a Markov generator.According to Proposition 2.6, if such a generator Q exists then it can be written as Log k 1 ,k 2 pAq for some k 1 , k 2 P Z.
Therefore, Lemma 5.6 implies that F pAq " ˆA1 0 0 A 2 ˙for some matrices A 1 and A 2 .Moreover, F pQq " ˆQ1 0 0 Q 2 ˙where Q 1 and Q 2 are real logarithms of A 1 and A 2 respectively.
As shown in the proof of Proposition 6.5, A 1 is actually a Markov matrix and Q 1 is a Markov generator for it (see also Lemma 5.6).Moreover, by Theorem 4 in [27], A 1 is embeddable if and only if LogpA 1 q or Log ´1pA 1 q are rate matrices.This implies that Log k 1 ,k 2 pAq is a rate matrix if and only if Log 0,k 2 pAq or Log ´1,k 2 are rate matrices.To conclude the proof we proceed as in the proof of Proposition 6.4.Indeed, note that for k P t0, ´1u, Log k,k 2 pAq " Log k,0 pAq `k2 V .Using this, it is immediate to check that Log k,k 2 pAq is a rate matrix if and only if N k " H and L k ď k 2 ď U k .

Discussion
The central symmetry is motivated by the complementarity between both strands of the DNA.When a nucleotide substitution occurs in one strand, there is also a substitution between the corresponding complementary nucleotides on the other strand.Therefore, working with centrosymmetric Markov matrices is the most general approach when considering both DNA strands.
In this paper, we have discussed the embedding problem for centrosymmetric Markov matrices.In Theorem 3.7, we have obtained a characterization of the embeddabilty of 4 ˆ4 centrosymmetric Markov matrices which are exactly the strand symmetric Markov matrices.In particular, we have also shown that if a 4 ˆ4 CS Markov matrix is embeddable, then any of its Markov generators is also a CS matrix.Furthermore, In Section 6, we have discussed the embeddability criteria for larger centrosymmetric matrices.
As a consequence of the characterization of Theorem 3.7, we have been able to compute and compare the volume of the embeddable 4 ˆ4 CS Markov matrices within some subspaces of 4 ˆ4 CS Markov matrices.These volume comparisons can be seen in Table 2 and Table 7.For larger matrices, using the results in Section 6, we have estimated the proportion of embeddable matrices within the set of all 6 ˆ6 centrosymmetric Markov matrices and within the subsets of DLC and DD matrices.This is summarized in Table 9 below.The computations were repeated several times obtaining results with small differences in the values but the same order of magnitude and starting digits.9: Relative volume of embeddable matrices within relevant subsets of 6 ˆ6 centrosymmetric Markov matrices.The results were obtained using the hit-and-miss Monte Carlo integration with 10 7 sample points.

Set
As we have seen in Section 3 and 6, we have only considered in detail the embeddability of CS Markov matrices of size n " 4 and n " 6.We expect that the proportion of the embeddable CS Markov matrices within the subset of Markov matrices in larger dimension tends to zero as n grows larger as indicated by Table 2, 7, 8, and 9.
These results together with the results obtained for the strand symmetric model (see Table 7) indicate that restricting to homogeneous Markov processes in continuous-time is a very strong restriction because non-embeddable matrices are discarded and their proportion is much larger than that of embeddable matrices.For instance, in the 2 ˆ2 case exactly 50% of the matrices are discarded [2, Table 5], while in the case of 4 ˆ4 matrices up to 98.26545% of the matrices are discarded (see Table 7) and in the case of 6 ˆ6 matrices the amount of discarded matrices is about 99.99863% as indicated in Table 9.However, when restricting to subsets of Markov matrices which are mathematically more meaningful in biological terms, such as DD or DLC matrices, the proportion of embeddable matrices is much higher so that we are discarding less matrices (e.g. for DD we discard 68.41679% of 4 ˆ4 matrices and 97.2441% of 6 ˆ6 matrices).This is not to say that it makes no sense to use continuous-time models but to highlight that one should take the above restrictions into consideration when working with these models.Conversely, when working with the whole set of Markov matrices one has to be aware that they might end up considering lots of non-meaningful matrices.
r and log pq ď log r.These inequalities are equivalent to the K3P-embeddability criteria presented in [37, Theorem 3.1] and [2,Theorem 1].Moreover, they are also equivalent to the restriction to centrosymmetricmatrices of the embeddability criteria for 4 ˆ4 Markov matrices with different eigenvalues given in[10, Theorem 1.1]

4 "
tpb, c, d, e, g, hq T P R 6 : b, c, d, e, g, h ě 0, 1 ´b ´c ´d ě 0, 1 ´e ´g ´h ě 0u.More explicitly, we identify the 4 ˆ4 CS Markov matrix ¨1 ´b ´c ´d b pb, c, d, e, g, hq P V M arkov 4 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25 a 31 a 32 a 33 a 32 a 31 a 25 a 24 a 23 a 22 a 21 a 15 a 14 a 13 a 12 a 11 12 a 13 a 14 a 15 a 16 a 21 a 22 a 23 a 24 a 25 a 26 a 31 a 32 a 33 a 34 a 35 a 36 a 36 a 35 a 34 a 33 a 32 a 31 a 26 a 25 a 24 a 23 a 22 a 21 a 16 a 15 a 14 a 13 a 12 a 11 M " ¨m11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 24 m 23 m 22 m 21 m 14 m 13 m 12 m 11 ‹ ‹ ‚ , where m 11 `m12 `m13 `m14 " 1 " m 21 `m22 `m23 `m24 and m ij ě 0. In biology, 4 ˆ4 centrosymmetric Markov (rate) matrices are often referred to as strand symmetric Markov (rate) matrices.In this article, we will use 4 ˆ4 centrosymmetric and strand symmetric interchangeably.Recall that the K3P matrices are assumed to have the form M " ¨m11 m 12 m 13 m 14 m 12 m 11 m 14 m 13 m 13 m 14 m 11 m 12 m 14 m 13 m 12 m 11 12 `b21 a 12 ´b21 a 11 ´b22 a 21 `b12 a 22 `b11 a 22 ´b11 a 21 ´b12 a 21 ´b12 a 22 ´b11 a 22 `b11 a 21 `b12 a 11 ´b22 a 12 ´b21 a 12 `b21 a 11 `b22 and only if A is a Markov matrix and |b 22 | ď a 11 , |b 21 | ď a 12 , |b 12 | ď a 21 , |b 11 | ď a 22 .iii) F ´1pC q is a rate matrix if and only if A is a rate matrix and b 22 ď a 11 pď 0q, |b 21 | ď a 12 p" ´a11 q, |b 12 | ď a 21 p" ´a22 q, b 11 ď a 22 pď 0q.
In this case, the only real logarithm of M 1 is the principal logarithm k 2 is a real matrix if and only if k 1 " k 2 " 0 and λ `µ ą 1.
k 4 are real.

Table 1 :
Number of samples in the sets V `, V em`, V DLC XV `, V DLC XV em`, V DD XV `and V DD XV emù sing hit-and-miss methods and Theorem 3.7.

Table 2 :
Relative volumes ratio between the relevant subsets obtained using hit-and-miss method and Theorem 3.7.The volumes were estimated as the quotient of the sample sizes in Table1.

Table 7 :
74 ˆ4 CS Markov matrices sampled uniformly.Embeddable matrices within 4 ˆ4 CS Markov matrices and its intersection with DLC matrices and DD matrices.
[23,osition 5.9.Let A P M n pRq be a CS matrix.If A is invertible, then it has infinitely many CS logarithms.Proof.The assumptions imply that the matrices A 1 and A 2 in Lemma 5.6 are invertible.By[23,  Theorem 1.28], each A 1 and A 2 has infinitely many logarithms.Hence, Theorem 5.8 implies that A has infinitely many centrosymmetric logarithms.
Proposition 5.10.Let A P M n pRq be a CS matrix such that LogpAq is well-defined.Then LogpAq is again centrosymmetric.
Sample points Embeddable sample points Rel.vol. of embeddable matrices V M arkov