Eigenértéke

Eigenértéke is a Hungarian term that translates to "eigenvalue" in English. In linear algebra, an eigenvalue is a scalar value that characterizes a linear transformation. When a linear transformation is applied to a vector, it can scale that vector by a specific factor. This scaling factor is the eigenvalue, and the corresponding vector is called an eigenvector.

Mathematically, for a square matrix A, a scalar $\lambda$ is an eigenvalue if there exists a non-zero vector v such that Av = $\lambda$v. Here, $\lambda$ represents the eigenvalue and v is the eigenvector. The equation Av = $\lambda$v can be rewritten as (A - $\lambda$I)v = 0, where I is the identity matrix. For a non-trivial solution (v $\neq$ 0), the matrix (A - $\lambda$I) must be singular, meaning its determinant is zero. This leads to the characteristic equation det(A - $\lambda$I) = 0, which is a polynomial in $\lambda$. The roots of this characteristic equation are the eigenvalues of the matrix A.

Eigenvalues and eigenvectors are fundamental concepts with broad applications in various fields. They are used in principal component analysis (PCA) for dimensionality reduction in data science, in quantum mechanics to describe the possible energy states of a system, in structural engineering to analyze the vibration modes of structures, and in image processing for tasks like face recognition. Understanding eigenértéke is crucial for comprehending these advanced mathematical and scientific applications.