Matrix bordering structure of the Faddeev–Jackiw algorithm: kernel reduction and symbolic automation

We show that the iterative Faddeev–Jackiw (FJ) reduction for singular Lagrangian systems constitutes a geometrically constrained instance of the Matrix Bordering Technique (MBT). For a first-order Lagrangian with singular pre-symplectic form, each iteration of the Barcelos–Neto–Wotzasek algorithm produces an extended symplectic matrix of canonical bordered form, f^(m) = ( array*20c f^(0) & B \\ - B^ & 0 \\ array ) f ( m ) = f ( 0 ) B - B ⊤ 0 where the bordering block B is determined by the gradients of the consistency constraints. Because f⁽⁰⁾ f ( 0 ) is singular by hypothesis, the classical Schur determinantal identity—which requires an invertible anchor block—cannot be invoked. The regularity of f⁽ᵐ⁾ f ( m ) is instead governed entirely by the interaction of B with (f⁽⁰⁾) ker ( f ( 0 ) ) . Writing N for an orthonormal basis of (f⁽⁰⁾) ker ( f ( 0 ) ) , we introduce the reduced constraint matrix Γ = N B Γ = N ⊤ B and prove the exact factorization (f⁽ᵐ⁾) = (∏ _ μ _ ² )\, (Γ )² det ( f ( m ) ) = ( ∏ ℓ μ ℓ 2 ) <