Section 5 1 Eigenvector and eigenvlaues

Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix.
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.
The concepts of eigenvectors and eigenvalues are very useful throughout pure and applied mathematics. Eigenvalues are also used to study difference equations and continuous dynamical systems. They provide critical information in engineering design, and they arise naturally in such fields as physics and chemistry.