Eigenväärtusel

Eigenväärtusel, in mathematics, refers to a fundamental concept in linear algebra related to linear transformations and matrices. When a linear transformation is applied to a specific non-zero vector, and the resulting vector is simply a scaled version of the original vector, then that scaling factor is called an eigenvaäärtusel. The corresponding non-zero vector is known as an eigenvector.

Formally, for a square matrix A, a non-zero vector v is an eigenvector of A if there exists a scalar λ such that Av = λv. In this equation, λ represents the eigenvaäärtusel associated with the eigenvector v. The eigenvaäärtusel indicates how the eigenvector is stretched or compressed by the linear transformation represented by the matrix A. If λ is positive, the eigenvector is scaled in the same direction. If λ is negative, the eigenvector is scaled and its direction is reversed. If λ is between 0 and 1, the vector is compressed, and if λ is greater than 1, it is stretched. If λ is 1, the eigenvector remains unchanged.

Eigenvaäärtusel and eigenvectors are crucial in various fields. They are used in principal component analysis (PCA) for dimensionality reduction, in quantum mechanics to describe observable quantities, in vibration analysis to determine natural frequencies, and in the study of dynamical systems to understand stability. Finding eigenvaäärtusel and eigenvectors involves solving the characteristic equation, which is derived from the condition det(A - λI) = 0, where I is the identity matrix and det denotes the determinant.

---