eigenértékhez

Eigenértékhez is a Hungarian term that translates to "eigenvalue" in English. In linear algebra, an eigenvalue is a scalar associated with a given linear transformation or square matrix. It represents a factor by which a corresponding eigenvector is stretched or shrunk when the linear transformation is applied.

For a square matrix A, a non-zero vector v is an eigenvector if applying the transformation A to v results in a scaled version of v. This relationship is expressed by the equation Av = λv, where λ is the eigenvalue and v is the eigenvector. The eigenvalue λ indicates the extent of the scaling. If λ is positive, the eigenvector is stretched in the same direction. If λ is negative, it's stretched in the opposite direction. If |λ| is greater than 1, there is an expansion; if |λ| is less than 1, there is a contraction. If λ is 0, the eigenvector is mapped to the zero vector.

Eigenvalues and eigenvectors are fundamental concepts with wide-ranging applications in various fields. They are crucial in understanding the behavior of systems described by linear transformations, such as in quantum mechanics, where they represent possible energy states of a system. In data analysis, particularly in techniques like Principal Component Analysis (PCA), eigenvalues and eigenvectors are used to identify the directions of greatest variance in data, enabling dimensionality reduction and feature extraction. They also appear in the study of differential equations, stability analysis of dynamical systems, and graph theory.