Scilab Function
Last update : 12/01/2004

projspec - spectral operators

### Calling Sequence

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

### Parameters

- `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`,

### Author

F. D.; ;