eigenvaäärtusel

Eigenvaäärtusel is a mathematical concept arising in linear algebra. It refers to a specific scalar value associated with a linear transformation, often represented by a matrix, that describes how certain vectors are scaled when the transformation is applied. These special vectors are known as eigenvectors.

When a linear transformation is applied to a vector, the vector generally changes its direction and magnitude. However, for certain non-zero vectors, the transformation only scales them. The factor by which the eigenvector is scaled is its corresponding eigenvaäärtusel. Mathematically, if A is a square matrix, v is a non-zero vector, and λ is a scalar, then Av = λv. In this equation, λ is the eigenvaäärtusel and v is the eigenvector.

Eigenvaäärtusel and eigenvectors are fundamental in various fields. They are used in analyzing the stability of dynamical systems, in principal component analysis for dimensionality reduction in data science, in quantum mechanics to describe observable quantities, and in Google's PageRank algorithm to rank web pages. Finding the eigenvaäärtusel of a matrix typically involves solving the characteristic equation, which is derived from the determinant of (A - λI) = 0, where I is the identity matrix. The roots of this polynomial equation are the eigenvaäärtusel.