Egenvektorene

Egenvektorene, or eigenvectors in English, are fundamental concepts in linear algebra associated with square matrices. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a vector that is a scalar multiple of itself. The scalar factor is known as the eigenvalue corresponding to that eigenvector.

Mathematically, for a square matrix A, a vector v (not equal to the zero vector) is an eigenvector if it satisfies the equation Av = λv, where λ is a scalar known as the eigenvalue. The eigenvalue indicates how much the eigenvector is stretched or compressed during the transformation represented by A.

Eigenvectors and eigenvalues are essential in various mathematical and applied fields, including differential equations, quantum mechanics, and principal component analysis. They help in understanding the properties of matrices, such as diagonalization, and facilitate the solution of systems of linear differential equations.

The process of finding eigenvalues involves solving the characteristic equation det(A - λI) = 0, where I is the identity matrix. Once the eigenvalues are determined, eigenvectors can be found by solving the associated linear equations (A - λI)v = 0.

Eigenvectors are unique up to scalar multiplication, meaning any scalar multiple of an eigenvector is also an eigenvector associated with the same eigenvalue. They form the basis for diagonalizing matrices and simplifying linear transformations, making them crucial for simplifying complex matrix operations and understanding linear systems' behavior.