feat(LinearAlgebra/Eigenspace/Matrix): a scalar is in the spectrum iff it's an eigenvalue

[SnirBroshi]

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#Is there code for X? > Matrix eigenvalues: spectrum and mulVec

[github-actions]

PR summary 6bdb86423a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff (regex)

+ hasEigenvalue_iff_exists_map_eq_smul
+ isUnit_iff_forall_mulVec_eq_zero
+ mem_spectrum_iff_exists_mulVec_eq_smul
+ mem_spectrum_of_map_eq_smul
+ mem_spectrum_of_mulVec_eq_smul

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.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 6bdb864).

• +5 new declarations
• −0 removed declarations

+Matrix.isUnit_iff_forall_mulVec_eq_zero
+Matrix.mem_spectrum_iff_exists_mulVec_eq_smul
+Matrix.mem_spectrum_of_mulVec_eq_smul
+Module.End.hasEigenvalue_iff_exists_map_eq_smul
+Module.End.mem_spectrum_of_map_eq_smul

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 6bdb86423a
Reference commit e2361c1beb

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[themathqueen]

This is a weird detour, going through spectrum to show this.

This is actually a one-liner, and doesn't need Field, only DivisonRing:

theorem isUnit_iff_forall_map_eq_zero [FiniteDimensional K V] {f : End K V} :
    IsUnit f ↔ ∀ v, f v = 0 → v = 0 :=
  f.isUnit_iff_ker_eq_bot.trans f.ker_eq_bot'

given that, I think just remove this? Or perhaps move to another PR if you think it's necessary?

[SnirBroshi]

Sure, but you were the one who asked me to add this in the first place

I think mem_spectrum_of_map_eq_smul is also unnecessary

[SnirBroshi]

Looking back at it maybe I misunderstood and you only asked for a linear map version of mem_spectrum_of_mulVec_eq_smul and not all three

[themathqueen]

Sorry, I should've been clearer. I think mem_spectrum_of_map_eq_smul could be useful (just like the matrix version)?

[themathqueen]

I don't think this belongs in this file now (not using anything about spectrum anymore)

[themathqueen]

to keep things consistent with the files, can you use \mu instead of a for the eigenvalue (here and below, and also for the matrix lemmas)?