With the advancement of data acquisition technologies, various formats and sources of datasets become available for different analysis purposes and application fields. This work aims at pushing statistical advances for better exploiting the information from multiple data subjects with a collective point of view, especially, through simultaneous diagonalization and joint object detection. In particular, for typical machine learning input data examples including time series, images, and videos, their structurally informative nature is often accompanied with underlying configuration patterns that may be of interest to a wide range of analysis. The main contribution of this work is to cleverly utilize the jointly uncovered information to introduce novel perspectives for data explorations and interpretations. This work consists of two main types of tasks. The first one, Simultaneous Diagonalization, is designed to study common eigen-patterns in multiple possibly asymmetric matrices. Motivated by various applications, a two-sample test as well as a generalization for multiple matrices are proposed. A partial version of the test is also studied to check whether a partial set of eigenvectors is shared across samples. Additionally, a novel algorithm for the considered testing methods is introduced. Simulation studies demonstrate favorable performance for all designs. Finally, the theoretical results are utilized to decouple multiple vector auto-regression (VAR) models into univariate time series, and to test for the same stationary distribution in recurrent Markov chains with different temporal resolutions. These applications are demonstrated using macroeconomic indices of 8 countries and streamflow discharge data, respectively. The second type of task focuses on Object Detection in images and is motivated by a solid material science application on Transmission Electron Microscopy (TEM) image processing. Spatially resolved \textit{in situ} TEM records the atom-scale dynamics with millisecond temporal resolution, which results in severely degraded signal-to-noise ratios (SNR). The poor SNR associated with high temporal resolution challenges the detection on atomic columns in catalyst materials undergoing structural dynamics. Two algorithms are developed sequentially to address the challenge. Blob Detection (BD) from the community of computer vision has been tailored to deal with TEM images of nanoparticle systems with high noise content present. It is demonstrated to outperform the results of other algorithms, enabling the determination of atomic column position and its intensity with a higher degree of precision, especially when SNR conditions are extremely severe. Other than classical frame-by-frame analysis, a ridge detection idea is developed as a natural extension of BD. This work harnesses temporal correlation across frames through simultaneous analysis of long image sequences, specified as a spatial-plus-temporal image tensor. Ridge detection is a classical tool to extract curvilinear features in image processing, and has great promise for trend filtering relatively stable atom-shaped objects in TEM videos. New ridge detection algorithms are defined to non-parametrically estimate explicit trajectories of atomic-level object locations as a continuous function of time. This approach is specially adapted to handle temporal analysis of objects that seemingly stochastically disappear and subsequently reappear throughout a sequence. The proposed method demonstrates high effectiveness and efficiency in simulation examples, and also delivers noticeable performance improvements in TEM implementations compared to benchmarks.