Eigenállapot

Eigenállapot, also known as an eigenvector or characteristic vector, is a fundamental concept in linear algebra. It represents a vector that, when a linear transformation is applied to it, does not change its direction. Instead, the vector is simply scaled by a factor, which is called the eigenvalue. Mathematically, this relationship is expressed as Av = λv, where A is the linear transformation (represented by a matrix), v is the eigenvector, and λ is the corresponding eigenvalue.

The significance of eigenvectors and eigenvalues lies in their ability to simplify the analysis of linear transformations. By decomposing a transformation into its constituent eigenvectors, one can understand how the transformation stretches or compresses space along specific directions. This is particularly useful in areas like quantum mechanics, where eigenállapot (or eigenstates) describe the possible states of a quantum system, and the corresponding eigenvalues represent the measurable quantities associated with those states, such as energy or momentum. In engineering, eigenvectors can indicate the modes of vibration of a structure, and eigenvalues can relate to the frequencies of these vibrations. The identification and calculation of eigenvectors and eigenvalues are crucial for solving systems of linear differential equations, performing principal component analysis in statistics, and understanding the stability of dynamical systems.