eigenértékelemzés

Eigenértékelemzés, or eigenvalue decomposition, is a fundamental concept in linear algebra. It involves breaking down a square matrix into a set of its eigenvectors and eigenvalues. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scaled version of itself. The scaling factor is known as the eigenvalue.

Mathematically, if A is a square matrix, v is an eigenvector, and λ is its corresponding eigenvalue, the relationship is expressed as Av = λv. This equation signifies that the linear transformation represented by matrix A simply scales the eigenvector v by the factor λ without changing its direction.

Eigenvalue decomposition is incredibly useful in various fields. In data science, it's a cornerstone of Principal Component Analysis (PCA), a technique used for dimensionality reduction by identifying the directions of greatest variance in data. In quantum mechanics, eigenvalues represent observable quantities like energy levels. Image compression, vibration analysis, and solving systems of differential equations also heavily rely on eigenértékelemzés.

The process of finding eigenvalues involves solving the characteristic equation det(A - λI) = 0, where I is the identity matrix and det denotes the determinant. Once the eigenvalues are found, the corresponding eigenvectors can be determined by solving the system of linear equations (A - λI)v = 0 for each eigenvalue λ. Not all matrices can be decomposed this way; they must typically be diagonalizable.