On the Block Triangular Form of Symmetric Matrices

Iain S. Duff (1, 2) , Bora Uçar (3, 4)

1 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
2 Numerical Analysis Group - STFC Rutherford Appleton Laboratory
3 GRAAL - Algorithms and Scheduling for Distributed Heterogeneous Platforms
4 LIP - Laboratoire de l'Informatique du Parallélisme

We present some observations on the block triangular form (btf) of structurally symmetric, square, sparse matrices. If the matrix is structurally rank deficient, its canonical btf has at least one underdetermined and one overdetermined block. We prove that these blocks are transposes of each other. We further prove that the square block of the canonical btf, if present, has a special fine structure. These findings help us recover symmetry around the anti-diagonal in the block triangular matrix. The uncovered symmetry helps us to permute the matrix in a special form which is symmetric along the main diagonal while exhibiting the blocks of the original btf. As the square block of the canonical btf has full structural rank, the observation relating to the square block applies to structurally nonsingular, square symmetric matrices as well.

Mots clés: sparse matrices; block triangular form; Dulmage Mendelsohn decomposition; Dulmage-Mendelsohn decomposition; maximum cardinality matchings

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Analyse numérique [cs.NA]

Iain S. Duff, Bora Uçar. On the Block Triangular Form of Symmetric Matrices. SIAM Review, 2010, 52 (3), pp.455--470. ⟨10.1137/080720036⟩. ⟨ensl-00536955⟩