Entrywise exponential of a PSD matrix is PSD

posSemidef_map_exp

Submitter: Kim Morrison.

Notes: Part of the Schur-Polya-Loewner theory of entrywise functions preserving PSD. The proof uses the Schur product theorem iteratively: exp_⊙(A) = ∑ A^{⊙k}/k!, each Hadamard power is PSD, and the convergent series of PSD matrices is PSD.

Source: I.J. Schoenberg, Positive definite functions on spheres, 1942.

Informal solution: Write exp(a_{ij}) as the convergent series ∑ (a_{ij})^k / k!. The matrix with entries (a_{ij})^k is the k-fold Hadamard product A^{⊙k}, which is PSD by iterated Schur product. The partial sums are nonneg combinations of PSD matrices, hence PSD. PSD is a closed condition, so the limit is PSD.

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n]
    {A : Matrix n n ℝ} (hA : A.PosSemidef) :
    (A.map Real.exp).PosSemidef := byn:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef
  sorryAll goals completed! 🐙

Solved by

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