projspec
========

spectral operators



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


::

    [S,P,D,i]=projspec(A)




Arguments
~~~~~~~~~

:A square matrix
: :S, P, D square matrices
: :i integer (index of the zero eigenvalue of `A`).
:



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

Spectral characteristics of `A` at 0.

`S` = reduced resolvent at 0 ( `S` = -Drazin_inverse( `A`)).

`P` = spectral projection at 0.

`D` = nilpotent operator at 0.

`index` = index of the 0 eigenvalue.

One has `(s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - S - s*S^2
-...` around the singularity s=0.



Examples
~~~~~~~~


::

    `deff`_('j=jdrn(n)','j=zeros(n,n);for k=1:n-1;j(k,k+1)=1;end')
    A=`sysdiag`_(jdrn(3),jdrn(2),`rand`_(2,2));X=`rand`_(7,7);
    A=X*A*`inv`_(X);
    [S,P,D,index]=projspec(A);
    index   //size of J-block
    `trace`_(P)  //sum of dimensions of J-blocks
    A*S-(`eye`_()-P)
    `norm`_(D^index,1)




See Also
~~~~~~~~


+ `coff`_ resolvent (cofactor method)


.. _coff: coff.html