Eigenvectors
Given a linear transformation A , a nonzero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation
 (1)
 for some scalarIn this situation, the scalar is called an "eigenvalue" of A corresponding to the eigenvectorIn other words the result of multiplying b y the matrix is just a scalar multiple of
The key equation in this definition is the eigenvalue equation, Most vectorswill not satisfy such an equation: a typical vector changes direction when acted on by A , so that is not a multiple ofThis means that only certain special vectors are eigenvectors, and only certain special scalars are eigenvalues. Of course, if A is a multiple of the unit matrix, then no vector changes direction, and all nonzero vectors are eigenvectors.
The requirement that the eigenvector be nonzero is imposed because the equation holds for every A and everySince the equation is always trivially true, it is not an interesting case. In contrast, an eigenvalue can be zero in a nontrivial way. Each eigenvector is associated with a specific eigenvalue. One eigenvalue can be associated with several or even with an infinite number of eigenvectors by scaling a vector.
acts to stretch the vector not change its direction, sois an eigenvector of A .
From (1) which we may factorise as hence det where I is the identity matrix.
 We may then form a polynomial equation in and solve it to find the eigenvalues:
 A= AλI= which becomesWe can simplify, factorise and solve.

 is an eigenvector.
is an eigenvector.