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
- 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 non-zero vectors are eigenvectors.
The requirement that the eigenvector be non-zero 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= A-λI=- which becomesWe can simplify, factorise and solve.
- is an eigenvector.
is an eigenvector.