Gausseiginleiðu

Gausseiginleiðu refers to the characteristic equation used in linear algebra to find the eigenvalues of a square matrix. It is derived from the definition of an eigenvalue and eigenvector, where for a matrix A and a non-zero vector v, Av = λv, and λ is the eigenvalue. Rearranging this equation gives Av - λv = 0, which can be written as (A - λI)v = 0, where I is the identity matrix. For a non-trivial solution (i.e., v ≠ 0), the matrix (A - λI) must be singular, meaning its determinant is zero. Therefore, the Gausseiginleiðu is the equation det(A - λI) = 0. Solving this polynomial equation for λ yields the eigenvalues of the matrix A. The term "Gausseiginleiðu" is an Icelandic term, with "Gausse" likely referring to Carl Friedrich Gauss, and "eiginleiðu" translating to characteristic equation. The fundamental theorem of algebra guarantees that this polynomial equation will have a number of roots (counting multiplicity) equal to the dimension of the matrix. These roots are the eigenvalues, which are crucial for understanding the behavior of linear transformations represented by the matrix, including stability analysis, principal component analysis, and solving systems of differential equations. The process of finding these eigenvalues is a cornerstone of many areas within mathematics, physics, engineering, and computer science.