Topics: Algebra - Eigenvalues and Eigenvectors


Let .

The polynomial is called the eigenpolynomial (polinomio característico).


Let and let be a basis for . We define the eigenpolynomial of T as:

…where . It’s common to find denoted as in books and other documentation.

We solve this polynomial find the eigenvalues of a given linear transformation.


The eigenpolynomial is a polynomial with degree that has as its main coefficient.


Let and with a vector space with finite dimension.

  1. The scalar is an eigenvalue of :
    • ( a basis for )
  2. has, at most, distinct eigenvalues; has, at most, distinct eigenvalues


Let with a vector space with finite dimension.

Let’s suppose that the characteristic polynomial of can be broken down into a product of degree 1 factors. Let be distinct eigenvalues of .

Then, the following propositions are equivalent:

  1. is diagonalisable
  2. If with , then
  3. If is the multiplicity of , then