Prove closed spectral conditions pass to matrix limits - #415
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Adds RealRooted/Mathlib/LinearAlgebra/Matrix/SpectrumClosed.lean: - `tendsto_det`: the determinant is continuous along entrywise-converging sequences (permutation expansion, algebra of limits); - `le_norm_eval_of_forall_le_norm_sub`: a monic complex polynomial whose roots avoid the epsilon-ball around mu satisfies `eps ^ natDegree <= ||p.eval mu||` (linear factorization over C); - `roots_mem_of_tendsto_eval`: roots of a limit polynomial stay in any closed set containing the roots along the sequence (constant degree, pointwise evaluation convergence) -- no multiset root topology, no Rouche: if all roots along the sequence avoided an epsilon-ball around a limit root mu, the evaluations at mu would be bounded below by eps^n while converging to zero; - `charpoly_roots_mem_of_tendsto`: the matrix form, with evaluation convergence supplied by `eval_charpoly` + `tendsto_det`; - `charpoly_roots_nonneg_real_of_tendsto`: **real nonnegative spectrum passes to entrywise limits** -- the root-continuity input for Gantmacher-Krein (#394). With this, full Gantmacher-Krein for totally nonnegative matrices reduces to a single remaining statement: every TN matrix is an entrywise limit of matrices with real nonnegative spectrum (Whitney density). Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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The "root continuity" input for #394, in a form that avoids multiset root
topology and Rouché entirely.
RealRooted/Mathlib/LinearAlgebra/Matrix/SpectrumClosed.lean:tendsto_det— determinant continuity along entrywise-converging sequences;le_norm_eval_of_forall_le_norm_sub— a monicp : ℂ[X]whose roots avoidthe
ε-ball aroundμhasε ^ natDegree ≤ ‖p.eval μ‖;roots_mem_of_tendsto_eval— roots of a limit polynomial stay in anyclosed set containing the roots along the sequence. Elementary
contradiction: bounded below by
εⁿvs converging to0;charpoly_roots_mem_of_tendsto— matrix form viaeval_charpoly;charpoly_roots_nonneg_real_of_tendsto— real nonnegative spectrum passesto entrywise limits of real matrices.
Consequence for #394: full Gantmacher–Krein now reduces to one remaining
statement — Whitney density (every TN matrix is an entrywise limit of matrices
with real nonnegative spectrum; TP or primitive-compound approximants suffice,
whose spectra are covered by #414).
Sorry-free; full library builds (9022 jobs).
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