Determinantal conjecture
Determinantal conjecture
Problem statement
determinantal_conjecture
Does the determinant of the sum of two normal complex matrices and always lie in the convex hull of the points ? Here the numbers and are the eigenvalues of and , and is an element of the symmetric group .
Lean 4
Formal statement
Lean 4/-
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-/
import FormalConjecturesUtil
/-!
# Determinantal conjecture
*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Determinantal_conjecture)
-/
/- Formalisation note: Here, we represent the two normal matrices as $U_1 D_1 U_1^*$ and
$U_2 D_2 U_2^*$ respectively, where $U_1, U_2$ are unitary and $D_1, D_2$
are diagonal. This is an universal form of a normal matrix,
allowing to retrieve $\lambda (A)_i$ as ${D_1}_{i,i}$, whereas the current mathlib
doesn't support obtaining the vector of eigenvalues from a general normal
non-Hermitian matrix. -/
namespace DeterminantalConjecture
/--
Does the determinant of the sum $A + B$ of two $n \times n$ normal
complex matrices $A$ and $B$ always lie in the convex hull
of the $n!$ points $\prod\_i (\lambda(A)\_i + \lambda(B)\_{\sigma(i)})$?
Here the numbers $\lambda(A)\_i$ and $\lambda(B)\_i$ are
the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric
group $S\_n$.
-/
@[category research open, AMS 15]
theorem determinantal_conjecture
(n : Type) [Fintype n] [DecidableEq n]
(d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :
(U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det
∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Perm n } := by
sorry
end DeterminantalConjecture
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
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