Sajátérték

Sajátérték is a Hungarian term that translates to "eigenvalue" in English. It is a fundamental concept in linear algebra and mathematics, particularly in the study of linear transformations and matrices. Eigenvalues are scalar quantities associated with a given linear transformation or matrix, and they reveal important properties about how the transformation affects certain vectors.

When a linear transformation is applied to a vector, the direction of the vector might change. However, there are specific, non-zero vectors, known as eigenvectors, that remain unchanged in direction after the transformation, only being scaled by a factor. This scaling factor is the eigenvalue corresponding to that eigenvector. Mathematically, for a square matrix A and a non-zero vector v, if Av = λv, then λ is the eigenvalue and v is the corresponding eigenvector.

Eigenvalues are crucial for understanding the behavior of systems described by linear equations, such as in physics, engineering, and computer science. For example, in quantum mechanics, eigenvalues represent possible measurable values of physical quantities. In principal component analysis, eigenvalues indicate the variance explained by each principal component, aiding in dimensionality reduction. They are also used in solving systems of differential equations and analyzing stability in dynamical systems. The set of all eigenvalues of a matrix is called its spectrum.