Abstract
Recorded electroencephalography (EEG) signals can be represented as square matrices, which have been extensively analyzed using mathematical methods to extract invaluable information concerning brain functions in terms of observed electrical potentials; such information is critical for diagnosing brain disorders. Several studies have revealed that certain such square matrices—in particular, those related to socalled “elementary EEG signals”—exhibit properties similar to those of prime numbers in which every square EEG matrix can be regarded as a composite of these signals. A new approach to ordering square matrices is pivotal to extending the idea of square matrices as composite numbers. In this paper, several ordering concepts are investigated and a new technique for ordering matrices is introduced. Finally, some properties of this matrix order are presented, and the potential applications of this technique to analyzing EEG signals are discussed.
1. Introduction and Motivation
Epilepsy is a common neurological disease that affects 1% of the world’s population [1]. People of all ages can be affected by this chronic brain disorder [2], which lowers sufferers’ quality of life with the possibility of a seizure occurring at any time. Early diagnosis can not only improve quality of life but also prevent patients from experiencing accidents. The diagnosis of epilepsy is often not straightforward, and misdiagnosis occasionally does occur [3]. A detailed and reliable eyewitness account of the event is the most crucial piece of information for indicative assessment, but this may not be accessible [4]. In most cases, electroencephalography (EEG) is an essential diagnostic test for assessing patients with possible epilepsy. Besides providing diagnostic support [5], EEG can also assist in classifying the underlying epileptic syndrome [6].
Mathematical analysis of EEG signals offers medical professionals vital information regarding brain activity during a seizure, thus increasing the understanding of complex brain function [7, 8]. In general, relevant information is extracted from EEG signals via two main methods: linear and nonlinear. Linear analysis (e.g., Fourier and wavelet transforms) has been successfully executed and has produced several good results [9–15]. However, since linear methods disregard the underlying nonlinear EEG dynamics, the results can only provide a limited amount of information concerning the brain’s electrical activity. By contrast, it is commonly accepted that the brain is a chaotic dynamical system; therefore, EEG signals generated by the brain are considered to be chaotic in another sense [16–18]. Additional information can be extracted from EEG signals by progressively incorporating a nonlinear analysis that reveals features that cannot be measured via linear methods [19].
Crucial to diagnosing the disorder is to solve the neuromagnetic inverse problem to identify the location of epileptic foci [20]. Therefore, fuzzy topographic topological mapping (FTTM) was introduced in [21] to determine the epileptic foci. More recently, FTTM has been extensively utilized to study the features of the recorded EEG signals of seizure patients (see [22–27]). Most notably, Yun in [28] claimed that one of the components of FTTM—namely, the magnetic contour ()—obeys the associative law, which is also satisfied in turn by events in time [29]. The author concluded by stating that the is a plane containing information. This prompted Binjadhnan in [25, 30] to perform Krohn–Rhodes decomposition on a set of square matrices of EEG signals, . The scholar found a remarkable result, namely, the EEG signals taken during an epileptic seizure (henceforth called EEGsignal square matrices for the remainder of this paper) are not chaotic, but rather exhibit ordered patterns in the form of simple algebraic structures, as expressed by Theorem 1.
Theorem 1 (see [30]). Any invertible square matrix of EEGsignal readings during an epileptic seizure at time can be written as a product of elementary EEG signals during an epileptic seizure in one and only one way.
Theorem 1 states that the elementary EEG signals (i.e., unipotent and diagonal EEG signals) constitute the building blocks of all EEG signals. This theorem, to a certain extent, is similar to the fundamental theorem of arithmetic, which holds that prime numbers are the multiplicative building blocks of the integers [31]. Equally significant are the results that indicate that has properties resembling those of prime numbers via the Jordan–Chevalley decomposition [32]. The wellordering property of positive integers is vital in producing one of the most beautiful results in the study of prime numbers, namely, the infinitude of prime numbers. Therefore, a technique of ordering matrices is required to extend the work of viewing the elementary EEG signals as prime numbers. The analogy of elementary EEG signals as prime numbers is of vital importance since the pattern of EEG signals can be investigated in terms of the pattern of prime numbers. Hence, the goal of this paper is to introduce a technique of ordering transformed EEG signals (in terms of square matrices).
The remainder of this paper is organized as follows. In Section 2, a brief review of a few concepts and techniques for ordering matrices is presented, along with their viability for ordering transformed EEG signals. In Section 3, a new technique for ordering matrices, namely, the precede operator, is introduced, allowing several ordering properties to be obtained. Next, this binary relationship is shown to fulfill the partialorder properties; beyond that, it is shown to be totally ordered in Section 4. In Section 5, several results are obtained when the order is applied to symmetric matrices. Then, the implementation of the precede operator to the real data of EEG signals is presented in Section 6. The interpretation of the results and their connection with the prime numbers are discussed in Section 7. Finally, we bring the paper to a close with concluding remarks concerning the need for such a partial order. Throughout the following sections, every matrix is considered to be a square matrix unless otherwise stated.
2. Concepts for Ordering Matrices
Over the past few decades, mathematicians and applied scientists alike have taken a deep interest in the ordering of matrices. Several order relations for matrix algebra have been produced in connection to a series of applications relevant to different branches of mathematics and its applications. These order relations include minus partial order [33], star partial order [34], sharp partial order [35], and matrix majorization [36]. Mitra et al. [37] wrote a comprehensive monograph in which they presented developments in the field of matrix ordering and shorted operators for finite matrices in a unified way, thus sparking research interest in this topic.
Matrix partial ordering has applications in many different areas; for instance, Liu [38] developed applications for comparing linear models. Moreover, in the field of statistics, Baksalary and Puntanen [39] presented the best linear unbiased estimator in a general GaussMarkov model. At the same time, Dahl et al. [40] characterized a binary relation involving stochastic matrices (namely, matrix majorization), which is very useful for comparisons of statistical experiments. Additionally, in the field of finance, Fontanari et al. [41] proposed a technique called quantum majorization to compare and rank correlation matrices such that portfolio risk can be more significantly assessed.
The minus partial order is the fundamental matrix partial order, of which almost all subsequent partial orders (including the star and sharp orders) constitute extensions. Such extensions have been created through the addition of restrictions to the minus partial order. The minus partial order (which was originally called the plus order) was established by Hartwig in [33] and independently by Nambooripad in [42] to generalize conventional partial orders on semigroups. Antezana et al. [43] and Šemrl [44] extended this partial order such that it could be applied in an objective way to operators on infinitedimensional spaces. Djikić et al. [45] documented a new representation of the minus order on the algebra of bounded linear operators on a Hilbert space. The natural partial order of Vagner on inverse semigroups and the star order of Drazin can be extended through minus order [37]. One key feature to note is that these partial orders are defined via the method of generalized inverses.
Another essential ordering concept for matrices is majorization, which has been applied across many fields including economics [46], statistics [47, 48], and, most recently, quantum mechanics [49]. This concept was first introduced in a classical book by Hardy et al. [50]. Later, Marshall et al. [36] extensively treated both the theory and application of majorization. Torgersen in [51, 52] studied the generalization of vector majorization and developed the theory of statisticalexperiment comparison. This theory is intended to answer the question “What conditions must be fulfilled in order to say that one statistical experiment provides more information than another?” A simple experiment can be found in [53], in which the conventional notion introduced by the author is closely related to that of vector majorization. However, while these generalizations evolved from statistical studies, they are not regularly discussed in the linearalgebra literature. Dahl [54] introduced and studied the generalization of (vector) majorization as it applies in the notable case of matrices with rows. The classical concept of majorization between vectors can be generalized via matrix majorization [55].
Some of the techniques of ordering matrices found in the literature, along with their realworld applications, advantages of the techniques, and their limitations (with respect to the purpose of ordering transformed EEG signals), are summarized in Table 1.
The techniques of ordering matrices summarized in Table 1 have been deemed unfit to be used to extend the work of Binjadhnan and Ahmad [25, 30] and Fuad and Ahmad [32] since there are some conditions required to be fulfilled, and some are limited to the special matrices. This offers the possibility of introducing and investigating a new partial order of square matrices as discussed in Section 3.
3. Precede Operator
As mentioned in Section 2, the set of square matrices of EEG signals during a seizure, , has properties similar to those of prime numbers. Therefore, EEGsignal square matrices can be assumed to be analogs of natural numbers. It can be said that one matrix is “greater” than another matrix, just as any natural number can be either greater than or less than another natural number since is a complete ordered field and . With this in mind, the precede operator, denoted by , is introduced as defined by Definition 1.
Definition 1. Let and be matrices and . Matrix is said to precede , written as , whenever the first exists for some . The comparison must be made in the sequence of rows, i.e., , until the first is discovered and denoted as . Otherwise, if , then . When , i.e., all the corresponding entries for each matrix are the same, then .
In other words, let us consider such thatDefine , such that and denote row in and , respectively.
Next, define such thatHence, matrix .
Now, define such thatIn other words, rearrange and reidentify the entries of .
Next, define such thatFinally, .
Definition 1 is introduced as a map between a matrix and a real number in , since is a complete ordered set.
Example 1. Consider two matrices and such thatAs can be clearly seen, the first is found. In this case, or . Then, .
Theorem 2. The mapping is welldefined.
Proof. Let , such that . Therefore, and imply thatHence, the mapping of is welldefined.
The execution of this definition can be summarized by Algorithm 1.
In short, , where is a composition of mappings such thatThe composition of mappings is best illustrated by Figure 1.

4. Ordered Matrices
In this section, several results are obtained to show that any square matrices together with the precede operator are ordered matrices.
Lemma 1. If , then .
Proof. Let . Then, there exists a first and a first for some such that . Suppose that is false; then, is true. Therefore, there exists a first and a first for some such that . This is impossible, since according to Definition 1, and it also contradicts with the assumption that says there exists a first and a first for some such that , as noted earlier.
Lemma 2. If and then .
Proof. Suppose ; then, there exists a first and a first for some , such that . Similarly, if then there exists a first and a first for some , such that . This is only possible if . In other words, , by Definition 1.
Lemma 3. If and , then .
Proof. Let . Then, there exists a first and a first for some such that . If , then there exists a first and a first for some such that . Suppose that is false; therefore, . In other words, there exists a first and a first for some , such that . There are three cases to consider:(i): if , then is a contradiction, since is no longer the first to be found, such that and , according to Definition 1.(ii): if , then is a contradiction, since is no longer the first to be found, such that but .(iii): if , then , since and , since . Therefore, , which immediately implies . In other words, , which is a contradiction .All three cases lead to contradictions; thus, if and , then .
Consequently, the binary relation is a partial order.
Theorem 3. The set is a partially ordered set.
Proof. The set is a partially ordered set since(i)It is reflexive, since by Definition 1(ii)By Lemma 2, is antisymmetric(iii)By Lemma 3, is transitiveOf equal importance, is totally ordered as well.
Theorem 4. The set is totally ordered.
Proof. According to Theorem 3, the set is partially ordered. Next, or . Consider the case where , and a contrapositive for Lemma 2 is applied. In other words, this means that . In summary, or or . Hence, is totally ordered.
5. Precede Ordering for Symmetric Matrices
When precede operator is applied to special matrices (namely, symmetric matrices), several properties are obtained.
Proposition 1. If and are symmetric matrices and , then (i.e., matrix transposition preserves for symmetric matrices).
Proof. Suppose that and are symmetric matrices and . It immediately follows that since and .
Proposition 2. If and are symmetric matrices and , then .
Proof. Suppose that ; then, there exists a first and a first for some . Nevertheless, , since for some , and and are the first terms of and respectively, that exhibit such conditions. Consequently, .
Proposition 3. If , then for (i.e., matrixpositivescalar multiplication preserves ).
Proof. Suppose that ; then, there exists a first and a first for some such that . Then, , since . Hence, .
Proposition 4. If and are skewsymmetric matrices and , then .
Proof. Suppose that and are skewsymmetric matrices; therefore, and . Nevertheless, if , then by Proposition 2. Consequently, , since and are skewsymmetric matrices.
Theorem 5. Suppose that and are positive symmetric matrices, such that and ; then, .
Proof. Suppose that , and are positive symmetric matrices. implies that there exists a first and a first for some , such that . Similarly, implies that there exists a first and a first for some , such that and a first for some , such that . Now, , since and are the fields. In short, the first the first for . This implies that .
Theorem 5 is best illustrated by Example 2.
Example 2. Consider the positive symmetric matrices , and such thatBy Definition 1, and is obtained. Next,Similarly, by Definition 1, is obtained.
Corollary 1. Suppose that and are positive symmetric matrices. If , then .
Proof. Suppose that and are positive symmetric matrices and that . Then, , according to Proposition 1. By replacing and in Theorem 5, we obtain .
Corollary 2. Suppose that and are positive symmetric matrices such that ; then, .
Proof. Suppose that and are positive symmetric matrices and . Then, , by Proposition 2. By replacing and in Theorem 5, we obtain .
Example 3. Suppose that and are two symmetric matrices, such thatThen, by Definition 1, Next,which, by Definition 1, implies that
Theorem 6. for every positive symmetric matrix (i.e., a positive symmetric matrix precedes its negation).
Proof. Suppose that and are positive symmetric matrices and that . Notice that, is a positive symmetric matrix sinceSimilarly, is also a positive symmetric matrix. Now, , according to Corollary 1, and , according to Proposition 2. Consequently, , by Theorem 5. Equivalently,Notice that if and are positive symmetric matrices, then is a symmetric matrix, since . Next, let and ; therefore, is symmetric. Replace in ; then, is obtained as required.
Example 4. Let be a symmetric matrix such thatThen, by Definition 1, we obtain .
Theorem 7. Suppose that and are positive symmetric matrices. If and , then .
Proof. Suppose that , and are positive symmetric matrices. Assume that and . According to Theorem 6, . Since and imply that by Lemma 3, we consequently obtain and by Theorem 3. In other words, . Therefore, by Proposition 3, which implies that .
Example 5. Suppose that , and are symmetric matrices such thatClearly, by Definition 1, . Next,
6. Implementation
As an example of the implementation of the precede operator on real data, two readings of EEG signals’ square matrices are presented. The EEG signals of epileptic seizure patients could be recorded and composed into a set of square matrices (see Figure 2). Firstly, the EEG data is gathered from the hospital by the EEG technologists and the threedimensional data are transformed into twodimensional data. The transformation of the EEG data into a lowerdimensional is executed via a novel technique called flattening the EEG, where the information can be preserved and conveniently analyzed [70].
The coordinate system of EEG signals, depicted in Figure 3(a), is defined as
(a)
(b)
Moreover, a function (see Figure 3(b)) is defined as
The mapping is an injective mapping of a conformal structure since both and were designed and proven as two manifolds [70]. Hence, the mapping can keep up the data in a specific angle and orientation of the surface throughout the recorded EEG signals. The technique of flat EEG is executed on three groups of EEG signals recorded from three different epileptic patients [70]. The author digitized the EEG signals during epileptic seizures at 256 samples per second using the Nicolet One EEG software. Next, each APD at every second was stored in a file that contained the position of an electrode on a magneticcontour () plane. Subsequently, the stored data were used to compose a set of square matrices.
Differences in surface potential could be recorded using an array of electrodes appended to the scalp; the computed voltages between pairs of electrodes are then clarified, amplified, and recorded. The most widely used system of electrode placement is the International TenTwenty System; this is a recommended standard method for characterizing the locations of electrodes at particular time intervals along with the head for recording scalp EEG [71]. The TenTwenty system depends upon the connection between the position of an electrode and the underlying area of the cerebral cortex (the “Ten” and “Twenty” refer to 10% and 20% interelectrode distances, respectively) [72]. The electrode position of this system is shown systematically in Figure 4.
(a)
(b)
Figure 4(a) illustrates the case where almost all of the electrodes are positioned or below from vertex . On the contrary, Figure 4(b) shows the electrode position from the top view of the head by modeling the head as a sphere. We assume that the hemisphere is from the top of the head [70]. In other words, from the front to the back is from to and from the left to the right is from to . In general, every APD at each second is stored in a file containing the position of electrodes on the plane, as tabulated in Table 2.
Then, the readings in Table 2 are rewritten in terms of a matrix, as shown below.
Let and a function be defined as
The mapping of can be rewritten as the following matrix:
Specifically,
The corresponding square matrix is generated by substituting the analogous average potential difference of every element into the above matrix. In particular, every single second of the APD is stored in a square matrix that contains the positions of electrodes on the plane.
Therefore, the plane becomes a set of matrices (EEG signals), which is written as follows:where is a potentialdifference reading for EEG signals from a particular sensor at time . For instance, the recorded EEG signals data during seizures at times and are tabulated in Tables 3 and 4.
The data in Tables 3 and 4 are then reordered in ascending order of values and tabulated in Tables 5 and 6, respectively, through the MATLAB programming developed by Binjadhnan [30]. Typically, the program exhibits twentyone sensor readings and four readings of zero (i.e., no recorded data from the “ghost” sensors at the specified time ); thus, a matrix is obtained from each program.
Table 5 yields the matrix:
Table 6 yields the matrix:
The two matrices, and , can be compared using the precede operator. Again,
Then, by Definition 1, the first that is greater than () is (i.e., ; hence, . Consequently, the precede operator can be used in the set of EEG signals’ square matrices, allowing one matrix to be said to precede another. The set of EEG signals’ square matrices, along with the precede operator, enables one to see the set of matrices as analogous to the set of real numbers, as related to each other with the “greater than,” operator. This analogy is best demonstrated in Figure 5.
7. Discussion
ptThe key advantage of using the precede operator over the techniques discussed in Section 2 is that it does not require to fulfill the necessary conditions (such as the existence of doubly stochastic matrix and generalized inverses), rather, the precede operator “inherits” the totallyordered property of real numbers. Thus, a similar result provides a piece of evidence that the elementary EEG signals during seizure contain similar attributes to the distribution of prime numbers among positive integers. The similarity of elementary EEG signals with prime numbers is summarized in Table 7.
The resemblance of elementary EEG signals with the prime numbers in terms of its ordering properties corroborates the assertions made by Binjadhnan [30], Barja [27], and Ahmad Fuad and Ahmad [74] that the EEG signals during seizures contain a similar pattern to that of the prime number distribution among positive integers. The premise of viewing the EEG signals as prime numbers enables one to study the dynamics of EEG signals during seizures in terms of the pattern of prime numbers. More importantly, the deduced pattern of seizures is critical to devise a methodology that is capable of predicting a seizure, which in turn would significantly improve the patients’ quality of life [75]. Conversely, it is instructive to explore the property of elementary EEG signals that could possibly exist in the distribution of prime numbers.
8. Conclusions
This study introduces a new technique for ordering square matrices, called precede operator, and this binary relation is proven to exhibit partial ordering. In addition, several results obtained by using this new ordering technique are presented. Furthermore, the newly introduced matrix partial order is applied to the square matrices of EEG signals, giving opportunities to further develop EEGsignal square matrices’ structure by studying those features that resemble some of the rich properties of prime numbers. In particular, it will be of interest to extend the totally ordered property of transformed EEG signals to the wellordered as well. We are working on this problem and hoping to present our findings in a future paper. Furthermore, as this paper is limited to the precede operator implemented only on the transformed EEG signals, it would be intriguing to investigate the feasibility of the proposed technique to other structures, such as, among others, the measurement of income inequality, statistical experiment, and thermodynamics, as previously summarized in Table 1.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors acknowledge the support of Universiti Teknologi Malaysia (UTM), Ministry of Higher Education Malaysia (MOHE), and Yayasan Sultan Iskandar (YSI). This research was also supported by Fundamental Research Grant Scheme FRGS PY/2020/05155 awarded by Ministry of Education.