  Solving linear systems and computing eigenvalues are two fundamental problems
in linear algebra. For solving linear systems, many efficient quantum
algorithms have been discovered. For computing eigenvalues, currently, we have
efficient quantum algorithms for Hermitian and unitary matrices. However, the
general case is far from fully understood. Combining quantum phase estimation,
quantum algorithm to solve linear differential equations and quantum singular
value estimation, we propose two quantum algorithms to compute the eigenvalues
of diagonalizable matrices that only have real eigenvalues and normal matrices.
The output of the quantum algorithms is a superposition of the eigenvalues and
the corresponding eigenvectors. The complexities are dominated by solving a
linear system of ODEs and performing quantum singular value estimation, which
usually can be solved efficiently in a quantum computer. In the special case
when the matrix $M$ is $s$-sparse, the complexity is $\widetilde{O}(s\rho^2
\kappa^2/\epsilon^2)$ for diagonalizable matrices that only have real
eigenvalues, and $\widetilde{O}(s\rho\|M\|_{\max} /\epsilon^2)$ for normal
matrices. Here $\rho$ is an upper bound of the eigenvalues, $\kappa$ is the
conditioning of the eigenvalue problem, and $\epsilon$ is the precision to
approximate the eigenvalues. We also extend the quantum algorithm to
diagonalizable matrices with complex eigenvalues under an extra assumption.

Quantum Physics;Numerical Analysis;Numerical Analysis