## 1 Introduction
The electrocardiogram (ECG) signals exhibit a complex behavior because of a range of physiological and pathological factors [17; 18; 19]. The analysis of the ECG signal is a complex task due to its intricate nature. Nevertheless, researchers are always involved in a proactive endeavor to develop intelligent, efficient, and effective methods for the purpose of clinical diagnosis [20]. Machine learning techniques have been extensively employed for the early detection of cardiac diseases. The utilization of deep learning algorithms is a prevalent practice in various domains, often employed for the purpose of executing tasks such as feature extraction and dimensionality reduction to extract pertinent information for clinical diagnosis and monitoring [21; 22]. In [23], a real-time deep learning model has combined the CNN and RNN for AF detection in long-term ECG recordings and has also achieved high sensitivity. Additionally, methods such as reconstructed phase space analysis [24], Lyapunov exponents [25; 26], correlation dimension [27; 28; 29], detrended fluctuation analysis (DFA) [30], recurrence plots and Poincare plots [31], among others, have been extensively researched to analyze the ECG signal [20]. [PERSON] et al. presented a noteworthy study that showcased a proficient mathematical methodology aimed at comprehending the complex relationship between RR interval and JT wave during a bicycle ergometry exercise [18]. Their research findings unveiled that the Perfect Matrices of Lagrange Differences (PMLD) is an effective matrix-based approach to examine the ECG signal and the self-organization of the cardiovascular system during the load and recovery phases of the stress test [18]. [PERSON] et al. also employed PMLD methodology to examine the RR interval and JT wave to gain insights into the mechanisms underlying the regulation of arterial blood pressure during the stress test [17]. [PERSON] et al., in another study, analyzed and proved that the PMLD can be employed for the early diagnosis of atrial fibrillation episodes [20]. It is worth mentioning that the utilization of matrix-based techniques has demonstrated efficacy even when working with datasets of limited size [20]. This paper draws inspiration from the works of [PERSON] et al. and [PERSON] et al., who have convincingly showcased the efficacy of a matrix-based approach for the analysis of ECG signals [17; 18; 20]. The main aims of this study are as follows:  
1. To present and analyze three different architectures of matrices for the early detection of atrial fibrillation episodes. The first two architectures are presented by and adapted from the work of [PERSON] et al. [32]. In this paper those matrices are labelled as Matrix Architecture One (MA1) and Matrix Architecture Two (MA2), respectively. It is important to mention that (MA1) and (MA2) are being used for the first time for the analysis of atrial fibrillation dataset. The third architecture is sourced from the work of the [PERSON] et al. [20], and is referred to as Perfect Matrices of Lagrange Differences (PMLDs). 2. It is noteworthy to mention and acknowledge that the groundwork for exploring the application of the PMLD for atrial fibrillation episodes has already been laid by [PERSON] et al. [20]. However, ref. [20] focuses on utilizing only three cardiac parameters of the recorded ECG--JT wave, QRS complex, and RR interval [20]. Building upon this foundation, the current paper aims to extend the application of the concept of PMLD matrices to employ a broader array of cardiac parameters. Thus, the multifold objectives of the study are summarized as follows: 1. To conduct a thorough analysis of the three matrix configurations to determine if the matrix norm or large discriminant yields favorable results for the analysis of ECG signals when utilizing only three cardiac parameters: JT wave, QRS complex, and RR interval. 2. Once the most suitable matrix architecture is identified, it will be subjected to additional analysis to evaluate its expandability.