Sajátvektorai

Sajátvektorai, which translates to "eigenvectors" in English, is a fundamental concept in linear algebra. Eigenvectors are special non-zero vectors that, when a linear transformation is applied to them, only change by a scalar factor. This scalar factor is known as the eigenvalue. In mathematical terms, for a square matrix A, a non-zero vector v is an eigenvector if Av = λv, where λ is the corresponding eigenvalue.

The concept of eigenvectors and eigenvalues is crucial for understanding the behavior of linear transformations. They reveal the directions in which a transformation acts purely by stretching or shrinking, without altering the vector's direction. This property makes them incredibly useful in various fields. For instance, in physics, eigenvectors can represent stable states or modes of vibration. In computer science, they are used in algorithms like principal component analysis (PCA) for dimensionality reduction and in Google's PageRank algorithm for ranking web pages.

Finding eigenvectors and eigenvalues involves solving a characteristic equation, which is derived from the definition Av = λv. This typically leads to finding the roots of a polynomial equation. The geometric interpretation is that eigenvectors represent the invariant directions of a linear transformation, and eigenvalues quantify the scaling factor along these directions.