On monotone Markov chains and properties of monotone matrix roots

: Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a speci ﬁ c subset of stochastic matrices. This article examines the embedding in a discrete - time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of 2 2 ( ) × monotone matrices are investigated in previous work. For 3 3 ( ) × monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all 3 3 ( ) × monotone matrices with positive eigen values have an m th root that satis ﬁ es the monotonicity conditions ( for all values m m , 2 (cid:2) ∈ ≥ ) . For monotone matrices of order n 3 > , diverse scenarios regarding the matrix roots are pointed out, and inter esting properties are discussed for block diagonal and diagonalizable monotone matrices.

Abstract: Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of 2 2 ( ) × monotone matrices are investigated in previous work. For 3 3 ( ) × monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all 3 3 ( ) × monotone matrices with positive eigenvalues have an mth root that satisfies the monotonicity conditions (for all values m m , 2 ∈ ≥ ). For monotone matrices of order n 3 > , diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.

Introduction
The embedding problem as introduced by Elfving in [1] investigates whether or not a discrete-time Markov chain is embeddable in a continuous-time Markov chain [2,3]. A Markov chain with a transition matrix P is continuous embeddable in case there exists a Markov generator G such that e P G = . Besides continuous embeddability, discrete embeddability is being studied. It regards the problem of whether or not a discrete-time Markov chain can be embedded in a discrete-time Markov chain [4,5]. Particularly, the discrete embedding problem involves the question of whether, for a given Markov chain regarding time unit 1, there exists a compatible Markov chain regarding time unit m 1 (m m , 2 ∈ ≥ ). The transition matrix of a Markov chain enables us to reformulate this problem: A Markov chain with a transition matrix P is discrete embeddable in case there exist a stochastic matrix A and a number m m , 2 ∈ ≥ satisfying P A m = . Such a matrix A is in fact an mth root of P within the set of stochastic matrices and is called a stochastic mth root of P. Several methods exist for computing matrix roots without focusing on stochastic roots [6,7]. Stochastic roots of n n ( ) × transition matrices are examined in some recent work [8,9], including studies that address in particular n 2 = and n 3 = [10,11]. In case for a transition matrix P there exists a stochastic root, we say that the matrix P is discrete embeddable within the set of stochastic matrices.
In building a Markov model, there are different possible situations in which one has to deal with a lack of data. One of these situations occurs when data are only available regarding time intervals that are greater than the time unit of the Markov model. In such a situation, the stochastic mth roots nevertheless provide insight into the transition probabilities regarding time intervals with length m 1 (m m , 2 ∈ ≥ ). In this way, a stochastic mth root enables us to find parameter estimations for a Markov chain in case there is a lack of appropriate data. In modeling a specific context, the transition matrix P of the Markov chain satisfies specific conditions. In examining the embeddability in such a situation, the existence of an arbitrary stochastic root is not satisfactory since the same specific conditions have to be fulfilled by the stochastic root of P. In this way, the transition matrix P as well as its matrix roots are both elements of the same specific subset of stochastic matrices. Recently, several articles have investigated matrix roots and the embedding problem within a specific subset of stochastic matrices: Matrix roots are examined within the set of symmetric and monomial matrices [12], within the set of circulant matrices [13], and within the set of state-wise monotone matrices [14]. The embedding problem for reversible Markov chains is investigated in [15] and for Kimura Markov matrices in [16].
Monotone Markov chains are introduced by Daley in [17] and further investigated by Keilson and Kester in [18]. Several authors point out the importance of monotone Markov models in practice for diverse contexts: for example, monotone Markov models for intergenerational occupational mobility, where the states are occupational categories that have a natural ranking from worst to best, and the transition probability p ij expresses the probability that a parent in state i will have a child in state j [19]. As well as monotone Markov models where the states are income classes arranged in an increasing order can be useful in modeling intergenerational income mobility [20]. Tests for stochastic monotonicity in intergenerational mobility tables are introduced in [21]. Monotone matrices also play an important role in equal-input modeling [22].
Monotone Markov models are useful in examining the evolution of credit ratings based on a credit rating transition matrix where the states are the ranked possible credit classes, with the first state being the highest rating AAA. A credit rating transition matrix should satisfy the monotonicity conditions [23]. The rating transition matrix is typically estimated for a 1-year period. One of the reasons for this is that the number of transitions within a shorter period is too small to result in a valid estimation for the transition probabilities [24]. In [24], the embedding problem in a continuous Markov chain for a credit rating transition matrix is examined. A square root of a yearly credit rating transition matrix would provide insights regarding the probability of the transition from rating i to rating j on semi-annual base. This example illustrates that mth roots of a monotone transition matrix provide useful insights. In case for a monotone transition matrix P there exists a monotone root, we say that the matrix P is discrete embeddable within the set of monotone matrices.
In this article, properties of the set of monotone matrices are examined. Furthermore, matrix roots and the discrete embedding problem are studied within this set. The article is organized as follows: Section 2 recalls some definitions and properties of the set of monotone matrices. In Section 3, matrix roots and the discrete embedding problem are discussed for n 2 = . Section 4 examines the monotonicity conditions for roots of 3 3 ( ) × monotone matrices. Section 5 discusses generalizations for n 3 > . The article concludes with avenues for future research in Section 6.

The set of monotone matrices
With regard to the eigenvalues of a monotone matrix, there are already some characteristics known: A monotone matrix is a stochastic matrix and has therefore always 1 as trivial eigenvalue. By introducing for , the nontrivial eigenvalue with largest modulus is always nonnegative [19].
A Markov generator satisfying equation (4) is called a monotone generator. In some situations, the problem of matrix roots of P can be examined via the Markov generator G since e G m ∕ is an mth root of P. Stochastic matrix roots and the general discrete embedding problem that examines the existence of mth roots of the transition matrix within the set n have been studied in previous work. Therefore, this article focuses on the question of whether for M MC n n n ∈ = ∩ , the monotonicity conditions are still satisfied for a matrix root of M. In the following sections, specific properties regarding matrix roots are examined more in detail for the particular cases of two-state and three-state monotone Markov chains.
3 Roots of ( × ) 2 2 monotone matrices A matrix P p ij 2 ( ) = ∈ has eigenvalues 1 and λ 0 ≥ according to equation (3). The case λ 0 = corresponds to an idempotent matrix P that has itself as mth root.
. The corresponding matrix roots P e G m m = ∕ are monotone since G is a monotone generator (according to equation (4)) and can be expressed as follows [10,25]: These insights brings us, in accordance with the result in [22], to the formulation of Theorem 1.
Theorem 1. Each monotone matrix P of order 2 2 ( ) × has P m (as defined in equation (5)) as monotone mth root (for all m m , 2 ∈ ≥ ). Each monotone Markov chain with two states is discrete embeddable and continuous embeddable within the set of monotone matrices.
One can verify that the monotone mth root P m , as introduced in equation (5), is the diagonal matrix with the eigenvalues 1 and λ as diagonal elements. The matrixT is a transformation matrix with columns that are right eigenvectors of P. Hence, P T D T 1 = × × − and , is the motivation to examine properties of the matrix roots P T D T 1 m m = × × − for diagonalizable matrices P of order n 2 > to find out whether or not Theorem 1 can be generalized.

Roots and embedding conditions for ( × ) 3 3 monotone matrices
The following theorem provides specific properties of the eigenvalues λ λ λ 1, and 1 2 3 = of a matrix P 3 ∈ as well as necessary embedding conditions. Theorem 2. For a monotone matrix P of order 3 3 ( ) × holds the following: The eigenvalues λ 1, 2 , and λ 3 of P are real-valued. If λ λ 2 3 ≥ , then λ 0 2 ≥ and λ λ 3 2 ≥ − . In case the matrix P is discrete embeddable within 3 holds λ Furthermore, in case P is discrete embeddable within 3 with an mth root for m 2 ≥ even number, both λ 2 and λ 3 are nonnegative.
Proof. Assume λ λ 2 3 | | | | ≥ . Then, according to [19], λ 2 is nonnegative. Consequently, the characteristic equation of P has λ 1 1 = and λ 2 as real-valued solutions and, therefore, also λ 3 ∈ . Furthermore, according to equation (3) holds that P λ λ tr 1 1 2 3 ( ) = + + ≥ . Hence, λ 0 2 ≥ and λ λ 3 2 ≥ − . In case the monotone matrix P is discrete embeddable within 3 , its eigenvalues are elements of Θ x y x x y x , 0 . 5 ; and therefore, additionally, λ 0.5 3 ≥ − . In case P is discrete embeddable within 3 , there exists a monotone matrix A 3 ∈ such that P A m = . Denoting the eigenvalues of A as μ μ 1, 1 2 = and μ 3 , the eigenvalues of P A m = are equal to μ 1, The eigenvalues λ 1 1 = and λ λ 2 3 ≥ of the monotone matrix P satisfy λ 0 2 ≥ . In case λ 3 is negative, the discussion of mth roots is, in accordance to Theorem 2, restricted to m odd. In other words, mth roots are examined for m λ 3 ( ) ∈ : A matrix P of 3 is a stochastic matrix with real-valued eigenvalues. Roots of 3 3 ( ) × stochastic matrices with real eigenvalues are studied in detail in [11], and those findings are therefore useful in what follows. The aim is now to investigate whether there exist matrix roots of P 3 ∈ that satisfy the monotonicity conditions. For M 3 ∈ , the conditions in equation (1)    with positive eigenvalues and minimal polynomial of degree 2 ≤ are diagonalizable. For these, and for all diagonalizable monotone matrix P more in general, the monotonicity properties of matrix roots are further examined in the following section.

Diagonalizable monotone matrices of order 3
For a diagonalizable matrix P 3 ∈ , there exists a transformation matrix T so that the diagonal matrix with diagonal elements λ 1, 2 and λ 3 . Let us denote, for all m is an mth root of P and has all row sums equal to 1 [11].
Expressing the diagonalizable matrices P and P m by their projections P P , 1 2 and P 3 results in the spectral decompositions: P P λ P λ P P P λ P λ P and .
Since both P and its mth root P m have the same projections, it has some advantages to reformulate the monotonicity conditions (6) based on the projections as in Lemma 4. The notation δ refers to Kronecker delta.
Proof. Since the projections satisfy A A A I is the identity matrix of order 3, the spectral decomposition of A A μ A μ A Moreover, the projection A 1 has all its rows equal and therefore: That is, in case μ In case μ μ 2 3 = , equation (7) results in: a a μ δ δ 0 In what follows, the monotonicity of the matrix root P m is examined for alternative possible scenarios for the eigenvalues λ 2 and λ 3 .
Lemma 5. For a monotone matrix P P λ P λ P Proof. According to Lemma 4, for P P λ P λ P In a similar way, for P P λ P λ P Since P is a monotone matrix, and therefore equation (8) holds, equation (9)  ≥ ≥ , the monotonicity conditions (6) also hold for the matrix root P P λ P λ P with nonnegative eigenvalues is discrete embeddable within 3 in case at least one of the roots P m is a stochastic matrix. This insight results in the following corollary. = + + be a diagonalizable monotone matrix with eigenvalues λ λ 0 2 3 ≥ > .
In case for n odd, the matrix root P P λ P λ P 1 2 2 3 3 n n n = + + satisfies the monotonicity conditions (6). Then all matrix roots P P λ P λ P For a diagonalizable monotone matrix P that has a negative eigenvalue, one can remark that in case the monotonicity properties (6) hold for the matrix root P m (with m a particular value in λ 3 ( )), this is automatically also the case for P 3 (as a result of Theorem 7). Therefore, if the third root P

Non-diagonalizable monotone matrices of order 3
For a non-diagonalizable matrix P, there exists a transformation matrix T such that P T J T 1 = × × − with the Jordan matrix J λ λ Consequently, the matrix P can be expressed as P P λP N λ λ 1 P TI T P  T I  I T N  TI T  , , ,

= + + with
where I ij denotes the 3 3 ( ) × matrix with i j , ( )th element equal to 1 and all the other elements equal to 0. For a monotone matrix P P λP N λ λ 1 = + + , the eigenvalue λ is nonnegative (according to equation (3)). In the case that λ 0 = there does not exist a stochastic root of P ( [11], Theorem 5). Hence, we know that a non-diagonalizable matrix P 3 ∈ with λ 0 = has no monotone matrix root. In the case that λ 0 > , is an mth root with all row sums equal to 1 [11], and this for all m \ 0 { } ∈ .
Theorem 9 proves, for a monotone matrix P P λP N λ λ 1 = + + with λ 0 > , that the monotonicity conditions are fulfilled for all mth roots P m . The following lemma provides some required properties beforehand. = + + ∈ with λ 0 > and P 1 , P λ , and N λ as in equation (10), the monotonicity conditions (6) also hold for the matrix roots P P λ P λ N Proof. Since the monotone matrix P satisfies p p  A monotone matrix is discrete embeddable within 3 if there exists a matrix root that satisfies the monotonicity conditions and that is simultaneously a stochastic matrix. In this way, the results in [11] ( Table 1), in combination with the results in Table 1 of this article provide full information on stochasticity as well as monotonicity of matrix roots for P 3 ∈ .

Further remarks and examples
In this section, some further remarks are formulated and examples are presented to highlight possible scenarios.
satisfies the monotonicity conditions. Moreover, this root is a stochastic matrix, and therefore, P has a monotone square root and is discrete embeddable within 3 .
has not all its elements nonnegative. In fact, the computation of all square roots of P let conclude that none of the square roots is a stochastic matrix and, therefore, neither a monotone matrix. One can remark that P has three distinct positive eigenvalues, and Theorem 3 is not applicable. In fact, having all eigenvalues positive is not a sufficient condition to be discrete embeddable within the set 3 . 5 Roots of ( × ) n n monotone matrices

Monotonicity conditions in matrix form
For higher order matrices, it has advantages to express the monotonicity conditions (2) more compact in matrix form. In a similar way, Conlisk defined in [19] necessary and sufficient conditions for a stochastic matrix to be a monotone matrix based on its dominance matrix. Let us introduce the n n 1 ( ( )) × − matrix S + with on its jth column the first jth elements equal to 0 and the other elements equal to 1, and the n n

Properties not generalizable from ≤ n 3 to > n 3
It is important to be aware that some properties, proven in Sections 3 and 4, no longer hold for n n ( ) × monotone matrices with n 3 > . Each 2 2 ( ) × stochastic matrix has all its eigenvalues real-valued. Hence, the same property holds for all matrices in 2 . Besides, a monotone matrix of order 3 3 ( ) × has all its eigenvalues real-valued according to Theorem 2. Nevertheless, this property does not hold any longer for higher order monotone matrices: does not satisfy the monotonicity conditions. The discrete embedding problem and investigating monotone matrix roots for general n n ( ) × monotone matrices with n 3 > is not easy. There are some alternative approaches to make progress in studying monotone matrix roots of n n ( ) × monotone matrices. In case a monotone matrix P is continuous embeddable within the set of stochastic matrices and a generator G does exist that satisfies equation (4), then P e G = is continuous embeddable within the set of monotone matrices. Hence, the matrix roots e G m ∕ are monotone and P is discrete embeddable within n . In applications, the transition matrix of a monotone Markov system under study is often the result of estimated transition probabilities based on an available dataset. Consequently, there is anyway a discrepancy between the estimated and the theoretical transition matrix. Therefore, if P is not continuous embeddable, then it is acceptable to replace P by an arbitrarily close approximation that is continuous embeddable [27] and that results in approximations for the roots of P, for which the monotonicity conditions can be examined.
Alternatively, since the set of diagonalizable monotone matrices is dense within n [28], a monotone transition matrix that is not diagonalizable can be approximated by an arbitrarily close diagonalizable matrix of n . Having insights regarding the properties of the roots of diagonalizable monotone matrices is then useful and that is where the following section focuses on.

Diagonalizable monotone matrices
In this section, we investigate roots and embedding conditions for diagonalizable monotone matrices of order n n ( ) × . , which proofs the lemma according to equation (12). □ From the discussion in Section 4, we know that, for a 3 3 ( ) × diagonalizable monotone matrix P with nonnegative eigenvalues, the roots P m satisfy the monotonicity conditions. On the other hand, the example in Fact 5.2 demonstrates that this property does not hold any longer for n 3 > . The question, therefore, is now under what conditions the result can be (partially) generalized to higher-order monotone matrices. Therefore, we consider a diagonalizable matrix P P λ P λ P n n n 1 2 2 = + +⋯+ ∈ , with nonnegative eigenvalues λ λ λ 1 0 according to equation (11). Hence, by using Lemma 10, the monotonicity conditions (2) can be expressed as follows: Since the matrix P P λ P λ P Moreover, the specificity of the identity matrix results in S I S I n n1 × × = − + − , and therefore, According to equation (15), in examining the monotonicity conditions, the sign of the function f x kl ( ) and, therefore, its zero points are of importance. Since the function f x kl ( ) is a sum of the exponential functions λ s x , the number of positive solutions of f x 0 kl ( ) = is at most equal to the number of sign changes in the sequence of coefficients in descending order of the basis λ λ n 2 ≥⋯≥ of the exponential functions [29]. Hence, for k l n , 1, , provides useful information regarding the number of positive zero points of the function f x kl ( ).
Proof. In case there is none or only one sign change in the sequence (18), then the function f x kl ( ) has at most one zero point in 0, 1 [ ]. Moreover, f 0 kl ( ) and f 1 kl ( ) are both nonnegative according to equation (16). .
In particular, f 0 kl m 1 ( ) ≥ and the theorem follows from equation (14). Consequently, one can conclude that none of the matrix roots P m satisfies the 1, 3 ( )th monotonicity condition, and therefore, none of the roots P m is a monotone matrix. ( ) ( ( )) ( ( )) = +… + ≥ . This means that the trace of all monotone matrices is at least equal to 1, and that for those that are block diagonal, the trace is even at least equal to the number of blocks. The following theorem formulates this characterization of the trace.

Block diagonal monotone matrices
Theorem 13. A block diagonal monotone matrix P satisfies P k tr( ) ≥ , where k is the number of blocks.

Further research questions
Within the set of monotone matrices, the discrete embedding problem is completely clarified for the case of 2 2 ( ) × matrices. Each monotone matrix is embeddable within the set 2 . Moreover, each 2 2 ( ) × monotone matrix has a monotone mth root, for all m \ 0 { } ∈ [22]. Within the set 3 holds, according to Theorems 6 and 9, that each monotone matrix with positive eigenvalues has an mth root that satisfies the monotonicity conditions. For n 3 > , roots within n are investigated for two important subsets of monotone matrices: for diagonalizable matrices in Section 5.3 and for block diagonal matrices in Section 5.4. For further research, it would be interesting to investigate matrix roots within n for more general n n ( ) × monotone matrices.