Seminar topic: Ultrametric matrices - multilevel structure, applications, inverses, eigenvalues
Ultrametric matrices are special class of symmetric, non-negative matrices where the entries
satisfy a strong form of the triangle inequality, speci cally related to hierarchical structures or
phylogenetic trees. They are often de ned by a tree structure.
Level structure: Ultrametric matrices have a special block 2 by 2 structure, where the o
diagonal blocks are constant rank one matrices, and the diagonal blocks (the next level) are again
ultrametric matrices. Such a structure is later used for H-matrices
Inverse Properties: The inverse of a strictly ultrametric matrix is a diagonally dominant
Stieltjes matrix (an M-matrix).
Tree Relation: Every ultrametric matrix corresponds to a weighted tree.
Applications: These matrices are frequently used in taxonomy (tree of life), molecular biology
(phylogenetic tree reconstruction), and modeling hierarchical latent concepts in data structure,
graph theory.
Nonsymmetric: Generalized ultrametric matrices.
Eigenvalues: The inverse structure is well-understood. But recently there is a huge interest
in eigenvalues of ultrametric matrices.
- Trainer/in: Reinhard Nabben