I need an example of a very special matrix.
- From: Carl Christian Kjelgaard Mikkelsen <cmikkels@xxxxxxxxxxxxx>
- Date: Wed, 30 Jan 2008 14:09:10 -0500
Hello.
I need an example of a real n by n matrix with some very specific properties.
Let P be the property
P: (r+s) is not 0 for all eigenvalues r and s.
(I am interested in P, because if a matrix real A satisfies P, then the Lyapunov equation AX+XA^T+Q=0 has a unique solution for every Q.)
Now what I would like is a matrix a real n by n matrix H, which is
1) upper Hessenberg and has property P,
2) each of the matrices H(1:k,1:k) does NOT have property P, for k < n.
Now, this is actually very easy: I realized that I could just use a matrix in companion form and this approach is wonderful, because it gives me complete control over the eigenvalues, in particular I can satisfy
3) the spectrum of H must be clustered around a few points preferably deep inside the left hand side of the complex plane.
However I want even more:
4) H must be diagonalizable and the condition number for the basis of eigenvectors much be small.
Now I can achieve 1,2, and 4 with the following example:
a=toeplitz([0 1 zeros(1,n-2)],[0 -1 zeros(1,n-2)]); a(n,n)=-1;
Unfortunately the spectrum is not very nice, but the matrix is nearly normal so the eigenvectors are nice. The matrix is stable, but the majority of the eigenvalues are concentrated close to +2i and -2i.
I have been unable to satisfy all four demands at the same time and I am not sure if they are even compatible.
Any ideas are most appreciated.
Carl Christian Mikkelsen.
.
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