DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Undirected structural graph of a Hermitian matrix

Definition

Let $A \in C^{n \times n}$ satisfy $A = A^{*}$ , where $A^{*} = {\overline{A}}^{T}$ . Define a real symmetric pattern matrix $H \in {0, 1}^{n \times n}$ by $H_{i i} = 0$ and, for $i < j$ , set $H_{i j} = H_{j i} = 1$ exactly when $A_{i j} \neq 0$ ; otherwise set both entries to zero. Since $A_{j i} = \overline{A_{i j}}$ , this pattern records exactly the undirected off-diagonal support of $A$ .

Apply the graph and elimination-ordering convention of Structural graphs and ordered symmetric elimination to $H$ . Its structural graph is the Hermitian structural graph of $A$ , and its filled graphs are the symbolic pattern used for Hermitian elimination. The real matrix $H$ records only support; numerical Schur complements are formed from $A$ .

Depends on

Structural graphs and ordered symmetric elimination

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jennifer Scott and Miroslav Tůma, Algorithms for Sparse Linear Systems, Chapter 4 (standard reference, not scraped)
Timothy A. Davis, Siva Rajamanickam, and Wissam M. Sid-Lakhdar, A survey of direct methods for sparse linear systems, §§4–5 (standard reference, not scraped)