How to find a permutation matrix to turn a general hermitian matrix into a block diagonal one?
12 vues (au cours des 30 derniers jours)
Afficher commentaires plus anciens
Huiyuan ZHENG
le 24 Avr 2022
Commenté : Huiyuan ZHENG
le 30 Juin 2022
If I have a hermitian matrix which satisfies H' = H, say H
H = [1, 0, -1i;
0, 1, 0;
1i, 0, 1],
is there a function to find a permutation matrix P so that
P'HP = [1, 0, 0;
0, 1, -1i;
0, 1i, 1]
a block diagonal one.
0 commentaires
Réponse acceptée
Christine Tobler
le 18 Mai 2022
First, we should keep in mind that the task is really to find a representation of A with as small blocks on the diagonal as possible. After all, we can always treat a matrix as block diagonal with one big block that's just the whole matrix.
One way you can think about this is to treat the matrix as an undirected graph: if A(i, j) is non-zero, there is an edge connecting node i and node j. We can then get the connected components of this graph (two nodes are in the same component only if there is a path connecting them). The nodes in each connected components represent the rows / columns of one diagonal block:
H = [1, 0, -1i;
0, 1, 0;
1i, 0, 1];
g = graph(H ~= 0);
plot(g)
nodeToComponent = conncomp(g) % nodeToComponent(i) gives component number of node i
Why is there this mapping? If we try to split up a connected component into smaller blocks, this isn't possible because there is always an edge connecting a node in proposed new component A with another node in proposed new component B. In terms of diagonal blocks, that would mean that H(i, j) ~=0 for i in proposed diagonal block A and j in proposed diagonal block B - so this couldn't be seen as a diagonal block.
You can get a permutation vector from
[nodeToComponentSorted, p] = sort(nodeToComponent)
H(p, p)
Plus de réponses (1)
Bruno Luong
le 24 Avr 2022
A = [1, 0, 1;
0, 1, 0;
1, 0, 1],
p=symrcm(A)
A(p,p)
3 commentaires
Bruno Luong
le 25 Avr 2022
Modifié(e) : Bruno Luong
le 25 Avr 2022
Your question is still not clear.
H is already block diagonal: a single block.
What is you criteria ? When I apply symrcm on you new small example of H, to me it is still fine:
H = [1, 0, -1i;
0, 1, 0;
1i, 0, 1];
p=symrcm(H);
Hp=H(p,p)
Hp-Hp' % still Hermitian?
% The below result shows that Hp have 2 Blocks: (2x2) and (1x1)
real(Hp)
imag(Hp)
Voir également
Catégories
En savoir plus sur Sparse Matrices dans Help Center et File Exchange
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!