LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Clearing dyadic row denominators preserves determinant sign

Statement

For every finite $s \times s$ matrix over dyadic rationals with $s \geq 1$ , multiplying each row by a suitable positive power of two gives an integer matrix whose determinant has the same sign and is zero exactly when the original determinant is zero.

Facts & Assumptions

Given: A finite $s \times s$ matrix $M$ with dyadic entries, and the Leibniz definition of determinant from Dyadic orientation and lifted sphere determinants.

[F1]

Dyadic floating-point format and directed rounding: an exact dyadic rational is $m 2^{e}$ for integers $m, e$ .

[F2]

Dyadic orientation and lifted sphere determinants: the determinant is the finite signed sum over permutations, with one entry from each row and column.

Proof

1.1givenF1choosealgebra

For each row $i$ , write its entries as $M_{i j} = m_{i j} 2^{e_{i j}}$ using [F1], and choose $k_{i} = \max (0, - \min_{j} e_{i j})$ . The row is finite, so this integer exists; then every $N_{i j} = 2^{k_{i}} M_{i j}$ is an integer.

2.1givenstep 1.1F2algebra

In every Leibniz term of $\det N$ , the product contains exactly one entry from each row, so it equals $(\prod_{i} 2^{k_{i}})$ times the corresponding term of $\det M$ . Summing the finite terms using [F2] gives $\det N = (\prod_{i} 2^{k_{i}}) \det M$ .

3.1step 2.1algebra∎

Every factor $2^{k_{i}}$ is positive. Therefore $\det N$ and $\det M$ have the same sign and vanish simultaneously.

Depends on

Used by

Dependency tree · two levels

4 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

Jonathan R. Shewchuk, Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, §4.2 (standard reference, not scraped)
CGAL Developer Manual, Robustness Issues, §11.1 (standard reference, not scraped)