Left eigenvectors using ARPACK

Question

I'm trying to find both the dominant $k$ left and right eigenvectors, that is,

$$V_L\mathcal{A} = \Lambda V_L\\ \mathcal{A}V_R = V_R\Lambda\\ V_LV_R = I_{k\times k}$$

• $V_L$ being the $k\times N$ matrix of left eigenvectors,
• $V_R$ being the $N\times k$ matrix of right eigenvectors, and
• $\Lambda$ being the diagonal $k\times k$ matrix of eigenvalues.

I'm using ARPACK (via scipy.sparse.linalg). scipy.sparse.eigs gives me the right eigensystem $V_R$ and $\Lambda$; I've tried two methods to get $V_L$:

1. 1. Apply scipy.sparse.linalg.eigs to $\mathcal{A}^T$ to get left eigenvectors and eigenvalues (call them $V_{L,0}$ and $\Lambda_L$)
   2. If $V_{L,0}V_R$ is singular or $\Lambda_L$ is too different from $\Lambda$, reduce $k$; if that doesn't work, restart
   3. Set $V_L \leftarrow (V_{L,0}V_R)^{-1}V_{L,0}$; or
   In principle this is reasonably fast (2x the running time of just getting the right eigenvectors), but in practice, in the cases where it needs to be restarted, it may need to re-run many times.
2. For each of the $k$ eigenvalues $\lambda_i$, apply scipy.sparse.linalg.lsmr to the overdetermined system $$\lambda_i^{-1}v_L^{(i)}\mathcal{A} = v_L,\\ v_L^{(i)}v_R^{(j)} = \delta_{i,j}$$ This is much more reliable than the other method, but it has a much worse best-case running time and it also has a quite limited accuracy.

I imagine it should in principle be possible to get left and right eigenvectors via IRAM; is this available in some way on ARPACK? (As mentioned I've been using scipy.sparse.linalg, but I'm amenable to accessing ARPACK more directly as a last resort). Is there some way to repackage this approximate eigendecomposition as something that ARPACK can handle (sort of like the $AA^\dagger$ trick for SVD)? Or is there some other way to get $V_L$, after having gotten $V_R$ and $\Lambda$, that's reasonably efficient, reliable, and accurate?

---

On the matrices I'm trying to diagonalize:

• Complex, non-Hermitian (also not symmetric, unitary, normal, etc.)
• Represented via scipy.sparse.linalg.LinearOperator, not technically a sparse matrix
• Equally efficient to perform left- and right- vector multiplication
• Magnitude of eigenvalues should theoretically decay fast (between polynomially and exponentially); initial eigenvalues tend to decay even faster
• For an example see arXiv:1510.00689; operator is defined in Fig. 2 and eigenvalue problem is described in Fig. A.4.b.