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Eigenvalues and Eigenvectors.

Let be an endomorphism of the vector space V.

Definition 11.4.1   A scalar for which there exists a non-zero vector such that is called an eigenvalue of and the vector is called an eigenvector of .

Example 11.4.2   If is defined by , then every vector is an eigenvector corresponding to the eigenvalue 2.

Example 11.4.3   is the endomorphism of whose matrix w.r.t. the standard basis is .

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:

• for , the eigenvectors are the vectors of the form , with .
• for , the eigenvectors are the vectors of the form , with .

Example 11.4.4   Let be the space of all functions from to which are infinitely many times derivable. Consider the endomorphism of the vector space V. We have and , i.e. the sine function and the cosine function are eigenvectors with eigenvalue -1 of the endomorphism .

Properties:

1.
Let be an endomorphism of the vector space V. If is an eigenvalue of , then the set of all the vectors with eigenvalue is a vector subspace of V.
2.
For , .
3.
If are eigenvectors of corresponding to the respective (distinct) eigenvalues , then the family is linearly independent.

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).

Next: Similar matrices. Up: Eigenvalues and eigenvectors. Previous: An application of Cayley-Hamilton's
Noah Dana-Picard
2001-02-26