Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a matrix is a non-zero vector that, when the matrix is multiplied by it, results in a scalar multiple of that vector. This scalar is known as the eigenvalue. Essentially, eigenvectors point in a direction that remains unchanged by the transformation represented by the matrix. These concepts are widely used in various fields, including physics, computer science, and data analysis. They help in simplifying complex problems, such as in principal component analysis (PCA), where they identify the most significant directions in data.