Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation

(1)

for some scalar In this situation, the scalar is called an "eigenvalue" of A corresponding to the eigenvector In other words the result of multiplying by the matrix is just a scalar multiple of

The key equation in this definition is the eigenvalue equation, Most vectors will not satisfy such an equation: a typical vector changes direction when acted on by A, so that is not a multiple of This 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 non-zero vectors are eigenvectors.

The requirement that the eigenvector be non-zero is imposed because the equation holds for every A and every Since 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, so is 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= A-λI=- which becomes We can simplify, factorise and solve.

is an eigenvector.

is an eigenvector.