What are the applications of Eigenvalue and eigenvector?

Eigenvalues and eigenvectors have numerous applications across various fields. In physics, they are used in quantum mechanics to describe states of systems; in engineering, they help analyze stability in structures and systems. In data science, they are crucial for techniques like Principal Component Analysis (PCA), which reduces dimensionality while retaining essential features. Additionally, they play a vital role in solving differential equations and are used in various algorithms in machine learning and computer vision.