feat(LinearAlgebra/Matrix): regularized inverse limit and trace continuity

[allenhaozhu]

Add Matrix.regularizedInv_mul_tendsto_one and the trace corollaries Matrix.trace_regularizedInv_mul_tendsto and Matrix.trace_regularizedInv_mul_tendsto_card: for a square matrix M over a normed field with det M ≠ 0, the Tikhonov-regularized expression (M + lam • 1)⁻¹ * M tends to the identity matrix as the scalar lam tends to zero, and consequently its trace tends to tr 1 = (Fintype.card d : R). The proofs compose NormedRing.inverse_continuousAt, continuousAt_matrix_inv, Matrix.nonsing_inv_mul, and Continuous.matrix_trace; no new definitions are added. These lemmas are useful in matrix-regularization limit arguments arising in numerical linear algebra (Tikhonov-regularized least squares, ridge regression) and in statistics.

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AI-assistance disclosure

Per the Mathlib AI-use policy:

• Tool. Claude Code (Anthropic) with the Claude Sonnet 4.6 model.
• Use. I specified the target lemma statements and the proof strategy; the assistant drafted Lean 4 tactic combinations against current Mathlib. I iterated on the proofs, verified each lemma builds under lake build from a clean checkout, and read the final code.
• Vouching. I have read every declaration in this file and can defend the proofs without further AI assistance. I welcome reviewer feedback on naming, namespace placement, and stylistic alignment with the surrounding Mathlib modules.

[github-actions]

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[github-actions]

PR summary f5f0fceeea

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.LinearAlgebra.Matrix.InverseLimit (new file)	1828

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Declarations diff

+ regularizedInv_mul_tendsto_one
+ trace_regularizedInv_mul_tendsto
+ trace_regularizedInv_mul_tendsto_card

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

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No changes to strong technical debt.

No changes to weak technical debt.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.