This paper presents quantum algorithms for estimating and transforming eigenvalues of high-dimensional non-normal matrices, which are not accessible via existing quantum singular value algorithms. The authors introduce two algorithms: Quantum EigenValue Estimation (QEVE) and Quantum EigenValue Transformation (QEVT). QEVE estimates eigenvalues of a diagonalizable matrix using a block encoding and a unitary preparing the corresponding eigenstate, achieving Heisenberg-limited scaling. QEVT transforms eigenvalues using Chebyshev and Faber approximations, providing nearly optimal query complexity. The algorithms are based on generating a Chebyshev history state through matrix generating functions, which encodes Chebyshev polynomials of the input matrix in quantum superposition. This state is used to estimate eigenvalues and apply polynomial transformations to the input matrix. The algorithms are applicable to a broad class of non-normal matrices with real spectra and Jordan forms, and can be extended to more general non-normal operators with eigenvalues in the complex plane. The results demonstrate that the proposed algorithms achieve optimal query complexity for eigenvalue estimation and transformation, and provide a unifying framework for processing eigenvalues of matrices on a quantum computer. The paper also discusses applications of these algorithms, including quantum differential equation algorithms and quantum ground state preparation.This paper presents quantum algorithms for estimating and transforming eigenvalues of high-dimensional non-normal matrices, which are not accessible via existing quantum singular value algorithms. The authors introduce two algorithms: Quantum EigenValue Estimation (QEVE) and Quantum EigenValue Transformation (QEVT). QEVE estimates eigenvalues of a diagonalizable matrix using a block encoding and a unitary preparing the corresponding eigenstate, achieving Heisenberg-limited scaling. QEVT transforms eigenvalues using Chebyshev and Faber approximations, providing nearly optimal query complexity. The algorithms are based on generating a Chebyshev history state through matrix generating functions, which encodes Chebyshev polynomials of the input matrix in quantum superposition. This state is used to estimate eigenvalues and apply polynomial transformations to the input matrix. The algorithms are applicable to a broad class of non-normal matrices with real spectra and Jordan forms, and can be extended to more general non-normal operators with eigenvalues in the complex plane. The results demonstrate that the proposed algorithms achieve optimal query complexity for eigenvalue estimation and transformation, and provide a unifying framework for processing eigenvalues of matrices on a quantum computer. The paper also discusses applications of these algorithms, including quantum differential equation algorithms and quantum ground state preparation.