We discover two complementary linear-combination-of-Hermitian-matrices (LCHM) formulations to achieve a general non-normal matrix eigenvalue transformation. Vanilla LCHM contains LCHS as a special case. Wely LCHM gives an exact formulation of matrix powers and polynomials without truncation and angular quadrature error.  LCHM formulas lead to new quantum eigenvalue transformation (QET) algorithms with optimal circuit depth, ancilla qubits, and postselection repetitions. LCHM-based QETs unify various quantum linear algebraic problems, including driven ODEs, iterative methods, resolvents, log functions, shifted fractional powers, Sign and ReLU transforms, and Faber approximation on non-circular domains. Besides optimal asymptotic scaling, Weyl LCHM with numerical-radius rescaling of weighted-shift matrices can exponentially reduce the prefactor in query complexity over prior QET methods. Reference: https://arxiv.org/abs/2607.25812

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