Eigenértékének

Eigenértékének is a Hungarian term that translates to "its eigenvalue" in English. It refers to a scalar value associated with a linear transformation of a vector space that is characteristic of that transformation. When a linear transformation is applied to a vector, and the result is a scalar multiple of the original vector, that scalar is called an eigenvalue. The corresponding vector is known as an eigenvector. Eigenvalues and eigenvectors are fundamental concepts in linear algebra with wide-ranging applications in various fields, including physics, engineering, computer science, and economics.

In mathematical terms, 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 can be expressed by the equation Av = λv, where λ is the eigenvalue corresponding to the eigenvector v. The eigenvalue λ represents the factor by which the eigenvector is stretched or shrunk by the linear transformation. If λ is positive, the eigenvector is stretched in the same direction. If λ is negative, it is stretched and its direction is reversed. If λ is zero, the eigenvector is mapped to the zero vector.

The concept of eigenvalues and eigenvectors is crucial for understanding the behavior of linear systems. For example, in quantum mechanics, eigenvalues represent observable quantities such as energy levels or momentum. In principal component analysis (PCA), a dimensionality reduction technique, eigenvalues indicate the variance of data along principal components, which are eigenvectors. The determination of eigenvalues typically involves solving the characteristic equation, which is derived from the determinant of (A - λI) = 0, where I is the identity matrix.