proofs
Provedmousseweighted-spectralmilestonesource-n1

Invertible left and right whitening give a bijection on rectangular updates

For all n,m∈N\htmlData{mx=1}{n, m \in \htmlData{mx=0}{\mathbb{N}}}, A,Ai∈Matrix⁡ Fin⁡ n Fin⁡ n R\htmlData{mx=9}{A, \mathit{Ai} \in \htmlData{mx=8}{\htmlData{mx=0}{\operatorname{Matrix}}\,\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{n}}\,\htmlData{mx=6}{\htmlData{mx=4}{\operatorname{Fin}}\,\htmlData{mx=5}{n}}\,\htmlData{mx=7}{\mathbb{R}}}} and B,Bi∈Matrix⁡ Fin⁡ m Fin⁡ m R\htmlData{mx=9}{B, \mathit{Bi} \in \htmlData{mx=8}{\htmlData{mx=0}{\operatorname{Matrix}}\,\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{m}}\,\htmlData{mx=6}{\htmlData{mx=4}{\operatorname{Fin}}\,\htmlData{mx=5}{m}}\,\htmlData{mx=7}{\mathbb{R}}}}, if A⋅Ai=1\htmlData{mx=2}{\htmlData{mx=0}{A} \cdot \htmlData{mx=1}{\mathit{Ai}}} \mathrel{\htmlData{mx=3}{=}} 1, Ai⋅A=1\htmlData{mx=2}{\htmlData{mx=0}{\mathit{Ai}} \cdot \htmlData{mx=1}{A}} \mathrel{\htmlData{mx=3}{=}} 1, B⋅Bi=1\htmlData{mx=2}{\htmlData{mx=0}{B} \cdot \htmlData{mx=1}{\mathit{Bi}}} \mathrel{\htmlData{mx=3}{=}} 1 and Bi⋅B=1\htmlData{mx=2}{\htmlData{mx=0}{\mathit{Bi}} \cdot \htmlData{mx=1}{B}} \mathrel{\htmlData{mx=3}{=}} 1, then

D↦A⋅D⋅B is bijective∧(∀D:Matrix⁡ Fin⁡ n Fin⁡ m R, Ai⋅(A⋅D⋅B)⋅Bi=D)∧(∀Y:Matrix⁡ Fin⁡ n Fin⁡ m R, A⋅(Ai⋅Y⋅Bi)⋅B=Y)\begin{aligned} & \htmlData{mx=16}{\htmlData{mx=15}{D \mapsto \htmlData{mx=5}{\htmlData{mx=3}{\htmlData{mx=1}{A} \cdot \htmlData{mx=2}{D}} \cdot \htmlData{mx=4}{B}}} \text{ is bijective}} \\ & \htmlData{mx=0}{\land} (\htmlData{mx=37}{\forall D : \htmlData{mx=36}{\htmlData{mx=28}{\operatorname{Matrix}}\,\htmlData{mx=31}{\htmlData{mx=29}{\operatorname{Fin}}\,\htmlData{mx=30}{n}}\,\htmlData{mx=34}{\htmlData{mx=32}{\operatorname{Fin}}\,\htmlData{mx=33}{m}}\,\htmlData{mx=35}{\mathbb{R}}},\ \htmlData{mx=25}{\htmlData{mx=23}{\htmlData{mx=17}{\mathit{Ai}} \cdot (\htmlData{mx=22}{\htmlData{mx=20}{\htmlData{mx=18}{A} \cdot \htmlData{mx=19}{D}} \cdot \htmlData{mx=21}{B}})} \cdot \htmlData{mx=24}{\mathit{Bi}}} \mathrel{\htmlData{mx=27}{=}} \htmlData{mx=26}{D}}) \\ & \htmlData{mx=0}{\land} (\htmlData{mx=58}{\forall Y : \htmlData{mx=57}{\htmlData{mx=49}{\operatorname{Matrix}}\,\htmlData{mx=52}{\htmlData{mx=50}{\operatorname{Fin}}\,\htmlData{mx=51}{n}}\,\htmlData{mx=55}{\htmlData{mx=53}{\operatorname{Fin}}\,\htmlData{mx=54}{m}}\,\htmlData{mx=56}{\mathbb{R}}},\ \htmlData{mx=46}{\htmlData{mx=44}{\htmlData{mx=38}{A} \cdot (\htmlData{mx=43}{\htmlData{mx=41}{\htmlData{mx=39}{\mathit{Ai}} \cdot \htmlData{mx=40}{Y}} \cdot \htmlData{mx=42}{\mathit{Bi}}})} \cdot \htmlData{mx=45}{B}} \mathrel{\htmlData{mx=48}{=}} \htmlData{mx=47}{Y}}) \end{aligned}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables
n, m range over natural numbers: the statement must hold for every choice. ((n m : ℕ))
Matrix
Matrix: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Matrix)
·Multiplication
A times Ai. (A * Ai)
=Equality
A · Ai and 1 are the same. (A * Ai = 1)
bijectiveBijective
D ↦ A · D · B pairs inputs and outputs one to one. (Function.Bijective (fun D : Matrix (Fin…)
Fin
Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application
Fin applied to n. Lean writes f x; papers write f(x). (Fin n)
x ↦ f(x)Function (λ-abstraction)
The function sending D to A · D · B. Lean writes fun x => …; papers write x ↦ …. (fun D : Matrix (Fin n) (Fin m) ℝ => A *…)
RReal numbers
The real numbers: every point on the number line. (ℝ)
∧And
Both hold: ∀ D, Ai · (A · D · B) · Bi = D, and ∀ Y, A · (Ai · Y · Bi) · B = Y. ((∀ D : Matrix (Fin n) (Fin m) ℝ, Ai * (…)
∀For all
The statement after the comma holds for every D. (∀ D : Matrix (Fin n) (Fin m) ℝ, Ai * (A…)

M1 / N1, Mousse arXiv:2603.09697v2 Section 3.2 pp. 4–5, equations (8)–(10). A,B are whitening factors; Ai,Bi are explicitly supplied two-sided inverses. Prove the sandwich map is bijective and both round-trip identities. Symmetry and positive definiteness are unnecessary for this algebraic component. The inverses are not inferred from positive semidefiniteness. This is the coordinate-map part of the paper's argument, not a convergence theorem or a proof of the polar minimum. Source record 01a0ebcb-ef77-7a6a-8d90-302eefa19141.

How it was proved

Proved
29 September 2026
By
Ajax
Checked by
Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms
Classical.choice, Quot.sound, propext
Proof
26 lines of Lean
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