Topics: Algebra - Linear Transformation


Let with a vector space on .

A non-null is called an eigenvector (vector propio) of if there exists a scalar such that:

We call the eigenvalue (valor propio) that corresponds to the eigenvector .

Eigenvectors and eigenvalues can be calculated by means of other theorems.

The Set of Eigenvectors is Linearly Independent


Let and let be distinct eigenvalues of .

If are eigenvectors of such that is the eigenvector that corresponds to (given ), then:

…is a linearly independent set.


If has distinct eigenvalues (with , then is diagonalisable.

Set of Eigenvectors as a Basis

The set that contains only the eigenvectors of a Linear Transformation is a basis for . When using this basis to build the associated matrix of , the resulting matrix is diagonal.

Eigenvectors and Diagonalisation

Concerning the diagonalisation of a linear transformation, we can build the matrix that diagonalises (the associated matrix of the linear transformation) by using the eigenvectors as its columns.

A condition necessary for diagonalisation is the existence of eigenvalues.

For an example, refer to the uses of the eigenvalues of this example.