Using SVD to biorthogonalize left and right eigenvectors?

Question

I have a set of left and right eigenvectors from an nonsymmetric eigenproblem, and I'd like to biorthogonalize them.

I tried Gram-Schmidt, but this fails for most cases.

I then read that the SVD is the best way to get an orthonormal basis for a matrix, where U would be my basis.

How can I extend the SVD to the case of two sets of eigenvectors? Or is there a better way to biorthogonalize my left and right eigenvectors?

EDIT: More details

I have a complex, non-Hermitian matrix,

$\qquad\mathbf{M} = \left[\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ -\mathbf{B^*} & -\mathbf{A^*}\end{array}\right]$

Where $*$ indicates conjugate transpose. I solve via ZGEEV for the left and right eigenvectors, so I have

$\qquad\mathbf{MX_R} = \mathbf{X_RE}\qquad$ and $\qquad \mathbf{X_LM} = \mathbf{EX_L}\qquad$

Where $\mathbf{E}$ is a diagonal matrix of eigenvalues, and $\mathbf{X_L}$ and $\mathbf{X_R}$ are matrices containing my left and right eigenvectors, respectively.

I can easily orthogonalize the right and left eigenvectors among themselves, i.e. $\mathbf{X_R^*}\cdot\mathbf{X_R} = \mathbf{I}$ or $\mathbf{X_L^*}\cdot\mathbf{X_L} = \mathbf{I}$, but what I want is to biorthonormalize them to each other, e.g. obtain:

$\qquad\mathbf{X_L}\cdot\mathbf{X_R} = \mathbf{I}$

I have written my own modified Gram-Schmidt to accomplish this, but like I said, it does not work for many cases. I cannot see any way of using LAPACK routines (i.e. ZGEQRF followed by ZUNGQR) to accomplish this, nor does using the SVD seem apparent to me.

Answer 1

I guess, the final result is suffering from instability (relative) of Gram-Schmidt, even in modified/stabilized forms. In this case, instead of going from the start: modifying the orthogonalization process, we can start with the final result instead.

So, after solving a non-Hermitian eigenvalue problem (with a reliable algorithm), we obtained $X_L$ and $X_R$, matrices of left and right eigenvectors, respectively.

We desire $$X_LX_R=I$$ however, $$X_LX_R=Y$$ where matrix $Y$ is unfortunately different from the identity $I$. Now, we can compute the LU decomposition (which is supposed to be much more numerically stable than Gram-Schmidt) for $Y=L_YU_Y$. $$ X_LX_R=L_YU_Y $$

Now, by solving two triangular systems with multiple RHSs via forward- (for $L_Y$) and back-substitution (for $U_Y$): $$ L_YX_L^\prime=X_L\\ X_R^\prime U_Y = X_R $$ we can solve for the biorthogonalized $X_L^\prime$ and $X_R^\prime$ via an also relatively stable numerical process.

So, the algorithm (definitely not the best in terms of computational complexity):

1. Compute $Y=X_LX_R$
2. Compute LU-decomposition of $Y$
3. Solve 2 triangular systems with multiple RHSs to obtain the final result of biorthogonolized matrices.

There is a small question of what happens if a set of eigenvectors is degenerate, but I don't see why LU should fail in that case either.