DefinitionDefinition: Literature-sourcedProof: 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.

Diagonal, lower, and upper parts in the stationary-iteration convention

Definition

Let $F$ be either $R$ or $C$ , let $n \geq 1$ , fix the coordinate order $1, \dots, n$ , and let $A = (a_{i j}) \in M_{n} (F)$ . Define $D, L, U \in M_{n} (F)$ by

$D_{i j} = {\begin{cases} a_{i i} & i = j, \\ 0 & i \neq j, \end{cases} L_{i j} = {\begin{cases} - a_{i j} & i > j, \\ 0 & i \leq j, \end{cases} U_{i j} = {\begin{cases} - a_{i j} & i < j, \\ 0 & i \geq j . \end{cases}$

Thus $D$ is diagonal, $L$ is strictly lower triangular, $U$ is strictly upper triangular (in the sense of Upper triangular, lower triangular and diagonal square matrices over a commutative ring), and $A = D - L - U$ . If $A = A^{*}$ , then $U = L^{*}$ , where $*$ denotes conjugate transpose. The decomposition uses the fixed coordinate order and is an exact-arithmetic identity.

Depends on

Upper triangular, lower triangular and diagonal square matrices over a commutative ring

Used by

Dependency tree · two levels

2 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

Yousef Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., §4.1 (standard reference, not scraped)
David Bindel, Matrix Computations (CS 6210), Cornell University, Fall 2012, Lecture 31 (standard reference, not scraped)