projspec

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 = blockdiag(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)