Let be an endomorphism of the vector space V.
This system of equations has a non trivial solution if, and only if, its determinant is equal to 0.
The endomorphism has two eigenvalues: 5 and -5.
Let's look for eigenvectors. We replace successively by 5 and by -5 in the last system and solve for (x,y). We have:
The case of a finite dimensional vector space:
V is now an n-dimensional vector space and a basis for V is given. To every endomorphism we associate a square matrix of order n. The eigenvalues of are the roots of the characteristic polynomial of ; they will be called the eigenvalues of the matrix . By the same way the eigenvectors of will ce also called the eigenvectors of .
Note that the previous results are independent of the choice of the basis (v.i. 5.2).