Description: An eigenvector is a fundamental concept in linear algebra that refers to a non-zero vector that, when transformed by a matrix, results in a vector that is a scalar multiple of the original vector. This means that the direction of the eigenvector remains unchanged, although its magnitude may change. Mathematically, if A is a matrix and v is an eigenvector, the relationship Av = λv holds, where λ is a scalar known as the eigenvalue. Eigenvectors are crucial for understanding the properties of linear transformations and have applications in various fields, from physics to statistics. Their study allows for the decomposition of matrices into simpler components, facilitating the solution of systems of linear equations and the understanding of data structure in multivariate analysis. Additionally, eigenvectors are essential in the diagonalization of matrices, simplifying calculations in various mathematical and scientific applications. In the context of various computational models, eigenvectors can help understand the dynamics of transformations applied to data sequences, providing insights into stability and long-term behavior.
History: The concept of eigenvector was introduced in the 19th century, in the context of the development of linear algebra. Mathematicians such as Augustin-Louis Cauchy and Hermann Grassmann contributed to the formalization of these concepts. However, it was the work of David Hilbert and others in the 20th century that consolidated the use of eigenvectors in various disciplines, including quantum mechanics and matrix theory.
Uses: Eigenvectors are used in various applications, such as quantum mechanics, where they describe states of physical systems, and in data analysis, where they help reduce dimensionality through techniques like Principal Component Analysis (PCA). They are also fundamental in control theory and the stability of dynamic systems.
Examples: A practical example of eigenvectors is found in Principal Component Analysis (PCA), where they are used to identify the main directions of variation in a dataset. Another example is in quantum mechanics, where the eigenvectors of an operator represent possible measurable states of a system.
