New DERIVE Instructions concerning Eigenvalues and Eigenvectors

Pablo Roa and A. de la Villa

Universidad Pontificia Comillas. ETSII (ICAI)
Departamento de Matematica Aplicada
Alberto Aguilera, 23 28015. Madrid, Spain

roaprietop@gmail.com, avilla@upcomillas.es


                                  Abstract:


In Engineering Schools in Spain on the topic of Linear Algebra, it is very
common to study, in a general way, eigenvalues and eigenvectors.

With the current DERIVE instructions, it is necessary to make the process of
calculation of the canonical Jordan form J and the corresponding matrix P,
associated with the matrix A, step by step.  In addition, if the matrix A is
not diagonalizable, the process to find the invariant generalized subspaces is
tedious.  For that reason, we have implemented some DERIVE instructions that
allow, in quite general circumstances, to obtain the invariant generalized
subspaces associated with the eigenvalues, the corresponding canonical Jordan
form and also, through a single instruction, the calculation of the canonical
Jordan form J and the matrix P.  The limitations of the procedures can be of
two types: The impossibility of the exact calculation of the eigenvalues and
DERIVE limitations, if the matrix is of a high order.  For that reason, it is
also possible to work in approximated mode to get the canonical Jordan form and
the matrix P with very good results.

We hope that the new instructions will be a very didactical tool to enhance the
learning of our students.

In the presentation, several examples of the different situations will be
analysed.