eigenvärdet

Eigenvärdet is a fundamental concept in linear algebra, representing a scalar value associated with a linear transformation that characterizes how a vector is scaled when that transformation is applied. In simpler terms, for a given square matrix A, an eigenvector is a non-zero vector v that, when multiplied by A, results in a scalar multiple of itself. This scalar multiple is the eigenvärdet. Mathematically, this relationship is expressed as Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvärdet.

Eigenvärden and their corresponding eigenvectors reveal important properties of the linear transformation represented by the matrix. They are crucial in various fields, including physics, engineering, computer science, and economics. For instance, in quantum mechanics, eigenvärden represent observable quantities like energy levels, while eigenvectors represent the corresponding quantum states. In principal component analysis (PCA), eigenvärden indicate the variance explained by each principal component, aiding in dimensionality reduction. The process of finding eigenvärden involves solving the characteristic equation det(A - λI) = 0, where I is the identity matrix. The roots of this polynomial equation are the eigenvärden of the matrix A.