bdiag
=====

block diagonalization, generalized eigenvectors



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [Ab [,X [,bs]]]=bdiag(A [,rmax])




Arguments
~~~~~~~~~

:A real or complex square matrix
: :rmax real number
: :Ab real or complex square matrix
: :X real or complex non-singular matrix
: :bs vector of integers
:



Description
~~~~~~~~~~~


::

    [Ab [,X [,bs]]]=bdiag(A [,rmax])


performs the block-diagonalization of matrix `A`. bs gives the
structure of the blocks (respective sizes of the blocks). `X` is the
change of basis i.e `Ab = inv(X)*A*X`is block diagonal.

`rmax` controls the conditioning of `X`; the default value is the l1
norm of `A`.

To get a diagonal form (if it exists) choose a large value for `rmax`
( `rmax=1/%eps` for example). Generically (for real random A) the
blocks are (1x1) and (2x2) and `X` is the matrix of eigenvectors.



Examples
~~~~~~~~


::

    //Real case: 1x1 and 2x2 blocks
    a=`rand`_(5,5);[ab,x,bs]=bdiag(a);ab
    
    //Complex case: complex 1x1 blocks
    [ab,x,bs]=bdiag(a+%i*0);ab




See Also
~~~~~~~~


+ `schur`_ [ordered] Schur decomposition of matrix and pencils
+ `sylv`_ Sylvester equation.
+ `spec`_ eigenvalues of matrices and pencils


.. _schur: schur.html
.. _sylv: sylv.html
.. _spec: spec.html