eigivektoren

Eigenvektoren are special non-zero vectors that, when a linear transformation is applied to them, change only by a scalar factor. This scalar factor is called the eigenvalue. Mathematically, for a square matrix A, a non-zero vector v is an eigenvector if Av = λv, where λ is the eigenvalue. This equation signifies that the transformation A acting on v simply scales v by the factor λ, without changing its direction.

The concept of eigenvectors and eigenvalues is fundamental in various fields of mathematics, physics, and engineering. They provide insight into the behavior of linear transformations and can be used to simplify complex systems. For instance, in linear algebra, finding eigenvectors and eigenvalues can help diagonalize a matrix, which simplifies many calculations.

Eigenvectors are always associated with a specific eigenvalue. A single matrix can have multiple eigenvectors and corresponding eigenvalues. If v is an eigenvector, then any non-zero scalar multiple of v is also an eigenvector with the same eigenvalue. The set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector, forms a subspace known as the eigenspace. Understanding eigenvectors is crucial for analyzing stability in dynamical systems, solving differential equations, and in data analysis techniques like Principal Component Analysis (PCA).