ominaisvektorer

Ominaisvektorer, or eigenvectors, are fundamental concepts in linear algebra. An eigenvector of a square 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, this relationship is expressed as Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvalue.

Eigenvectors represent directions that are unchanged by the linear transformation defined by the matrix, only their magnitude is altered. This property makes them invaluable in various fields. For instance, in physics, eigenvectors can describe the principal axes of rotation or the modes of vibration of a system. In computer science, they are crucial for algorithms like principal component analysis (PCA) used in dimensionality reduction and image recognition, and for Google's PageRank algorithm that ranks web pages.

Finding eigenvectors and eigenvalues typically involves solving the characteristic equation, which is det(A - λI) = 0, where I is the identity matrix and det denotes the determinant. Once the eigenvalues are found, they can be substituted back into the equation (A - λI)v = 0 to solve for the corresponding eigenvectors. A matrix can have multiple eigenvalues and eigenvectors, and the set of all eigenvectors for a given eigenvalue forms an eigenspace.