Egenvektorer

Egenvektorer is a term used in the field of linear algebra to describe a set of orthogonal vectors that, when multiplied by a matrix, produce new vectors that are scaled by a scalar and then added to the original vector.

In mathematical terms, a set of egenvektorer can be defined as a non-trivial solution to the equation Ax = λx, where A is a matrix, x is the vector of egenvektorer, and λ is a scalar. This means that when the matrix A multiplies the vector x, the result is a scalar multiple of x, resulting in an eigenvector.

Egenvektorer have several important properties. They are orthogonal to each other, meaning that when you multiply one egenvektor by another, the result is zero. They also have a non-zero determinant, which ensures that the set of egenvektorer is linearly independent.

The discovery of egenvektorer is attributed to mathematician David Hilbert in the 1890s. He used them to study the theory of linear transformations and the canonical forms of matrices. Since then, egenvektorer have been widely used in various fields such as physics, engineering, and economics to study systems of linear equations.

One of the key uses of egenvektorer is to find the eigenvalues and eigenvectors of a matrix. This is a fundamental problem in linear algebra and has numerous applications in various fields. The process of finding egenvektorer is known as the eigenvalue decomposition, and it is a powerful tool for analyzing and solving systems of linear equations.

Overall, egenvektorer play a crucial role in understanding the properties and behavior of matrices, and their applications continue to expand into new fields of study.