Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts

We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A = A^T , which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product <u, v>_* = sum_i u_i v_i. This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered “prematurely” on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo–Hermitian and PT-symmetric Hamilton matrices. Numerical reference values are provided.

本描述了一项针对复对称矩阵（非厄米矩阵）的矩阵对角化算法，记为A = A^T。该算法基于两步法，包括基于非定内积＜u, v＞_* = ∑_i u_i v_i的广义豪斯霍尔德反射。此内积在两个参数上均呈线性，且避免了复共轭。通过广义豪斯霍尔德变换（第一步），将复对称输入矩阵转化为三对角形式。然后，通过使用隐式平移的迭代广义QL分解对三对角矩阵进行分解，使其收敛至对角形式（第二步）。当在超对角线或亚对角线上“过早”遇到机器精度零时，QL算法采用迭代降维技术。该算法允许对由复尺度哈密顿算子产生的共振和反共振能量进行可靠且计算高效的计算，以及对伪厄米和PT对称哈密顿矩阵的实能量本征值进行数值求解。同时，提供了数值参考值。