## Eigenvalues

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 &quot;eigenvalue&quot; of A corresponding to the eigenvector In 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 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.  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.  