A Direct Polynomial Approach to Spectral Decomposition

What the math gives to ML

The paper gives an explicit Hermite-interpolation construction of spectral projectors directly as matrix polynomials, including non-diagonalizable matrices with nontrivial nilpotent components. The transferable asset is that invariant spectral subspaces and matrix functions can be evaluated using only repeated matrix multiplications, avoiding an eigendecomposition at inference time and retaining information about Jordan structure. A practical target is a small state-space or recurrent layer whose transition operator is decomposed into polynomially computed modes, allowing mode-specific stability constraints, gains, dropout, or truncation. The main engineering limitation is numerical conditioning when eigenvalues are close, so an initial implementation should use normal or symmetric transition matrices and refresh the polynomial coefficients only occasionally.