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)