In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix.

They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of applied mathematics as diverse as finance and quantum mechanics. In general, a matrix acts on a vector by changing both its magnitude and its direction.

However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or, possibly, reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed.

This factor is the eigenvalue associated with that eigenvector. An eigenspace is the set of all eigenvectors that have the same eigenvalue. The concepts cannot be formally defined without prerequisites, including an understanding of matrices, vectors, and linear transformations.

The technical details are given below. In linear algebra, there are two kinds of objects: scalars, which are just numbers, and vectors, which can be thought of as arrows, and which have both magnitude and direction. In place of the ordinary functions of algebra, the most important functions in linear algebra are called "linear transformations", and a linear transformation is usually given by a "matrix", an array of numbers.

Thus instead of writing f(x) we write M(v) where M is a matrix and v is a vector. The rules for using a matrix to transform a vector are given in the article linear algebra.

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