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 discretetime monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of
1 Introduction
The embedding problem as introduced by Elfving in [1] investigates whether or not a discretetime Markov chain is embeddable in a continuoustime Markov chain [2,3]. A Markov chain with a transition matrix
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
In modeling a specific context, the transition matrix
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
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
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
2 The set of monotone matrices
A transition matrix of a Markov chain with
The concept of monotone matrix is introduced by Daley [17]. A monotone matrix is a stochastic matrix satisfying the following monotonicity conditions:
The set of
where
A monotone transition matrix is characterized by the fact that each row
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
With regard to the trace
This insight follows from using iteratively the monotonicity conditions (1):
It is well known that for
A monotone Markov chain with transition matrix
A Markov generator satisfying equation (4) is called a monotone generator. In some situations, the problem of matrix roots of
Stochastic matrix roots and the general discrete embedding problem that examines the existence of mth roots of the transition matrix within the set
3 Roots of
(
2
×
2
)
monotone matrices
A matrix
These insights brings us, in accordance with the result in [22], to the formulation of Theorem 1.
Theorem 1
Each monotone matrix
One can verify that the monotone
The result that
4 Roots and embedding conditions for
(
3
×
3
)
monotone matrices
The following theorem provides specific properties of the eigenvalues
Theorem 2
For a monotone matrix P of order
The eigenvalues
Proof
Assume
In case the monotone matrix
In case
The eigenvalues
A matrix
For a
Theorem 3
Each monotone matrix P of order
Matrices
4.1 Diagonalizable monotone matrices of order 3
For a diagonalizable matrix
Expressing the diagonalizable matrices
Since both
Lemma 4
For
Proof
Since the projections satisfy
That is, in case
In case
In what follows, the monotonicity of the matrix root
Lemma 5
For a monotone matrix
Proof
According to Lemma 4, for
In a similar way, for
Since
Theorem 6
For a diagonalizable monotone matrix
Proof
The configurations
In case
In case
Diagonalizable monotone matrices of order
Corollary 6.1
Each diagonalizable monotone matrix P of order
Theorem 7
Let
Proof
The eigenvalues of
For a diagonalizable monotone matrix
For example, for the monotone matrix
4.2 Nondiagonalizable monotone matrices of order 3
For a nondiagonalizable matrix
where
For a monotone matrix
Lemma 8
For a nondiagonalizable stochastic matrix
Proof
By definition of
Furthermore,
and also equivalent with
Theorem 9
For a nondiagonalizable monotone matrix
Proof
Since the monotone matrix
Furthermore,
Combining all the properties that are presented in this section leads to the conclusions summarized in Table 1, for
P diagonalizable 


Theorem 6  



Theorem 7  

No such monotone matrix


Equation (3)  
P nondiagonalizable 


Theorem 9  

No matrix root of


Theorem 5 in [11]  

No such monotone matrix


Equation (3) 
A monotone matrix is discrete embeddable within
4.3 Further remarks and examples
In this section, some further remarks are formulated and examples are presented to highlight possible scenarios.
Theorem 6 guarantees for a diagonalizable matrix
Fact 4.1. Although
For example, for
Fact 4.2.
Theorem 6 proves for a diagonalizable monotone matrix
For example, for the monotone matrix
5 Roots of
(
n
×
n
)
monotone matrices
5.1 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
Multiplying
Consequently,
One can remark, for
5.2 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
Each
Fact 5.1. A monotone matrix of order
For example,
For each
Fact 5.2. The monotonicity properties do not necessarily hold for
For example,
does not satisfy the monotonicity conditions.
The discrete embedding problem and investigating monotone matrix roots for general
In case a monotone matrix
Alternatively, since the set of diagonalizable monotone matrices is dense within
5.3 Diagonalizable monotone matrices
In this section, we investigate roots and embedding conditions for diagonalizable monotone matrices of order
Lemma 10
For a diagonalizable matrix
Proof
The diagonalizable matrix
From the discussion in Section 4, we know that, for a
By introducing for
we have that
according to equation (11).
Hence, by using Lemma 10, the monotonicity conditions (2) can be expressed as follows:
Since the matrix
Moreover, the specificity of the identity matrix results in
According to equation (15), in examining the monotonicity conditions, the sign of the function
Hence, for
provides useful information regarding the number of positive zero points of the function
Let us introduce the notations
in order to be able to formulate in Theorem 11 sufficient conditions that guarantee that the matrix root
Theorem 11
In case
Proof
In case
Theorem 12
In case for the diagonalizable monotone matrix P with projections
for
Proof
In case there is none or only one sign change in the sequence (18), then the function
The result of Theorem 12 is useful in studying the existence of monotone roots. For example, for
we have already mentioned in Section 5.2 that
one can note that the sequence (18) has two sign changes for
5.4 Block diagonal monotone matrices
In case of a block diagonal Markov model with
The nature of the monotonicity conditions (1) results for a block diagonal matrix
Therefore, for a monotone matrix
Theorem 13
A block diagonal monotone matrix P satisfies
Furthermore,
6 Further research questions
Within the set of monotone matrices, the discrete embedding problem is completely clarified for the case of

Conflict of interest: The author states no conflict of interest.

Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.
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