Purpose

LA_GEEVX computes for a real or complex square matrix $A$, the eigenvalues and, optionally, the left and/or right eigenvectors. Optionally, it also balances $A$ and computes reciprocal condition numbers for the eigenvalues and right eigenvectors.
A right eigenvector $v_j$ of $A$ satisfies

$$A \, v_j = \lambda_j \, v_j$$

where $\lambda_j$ is its eigenvalue. A left eigenvector $u_j$ of $A$ satisfies

$$u_j^H \, A = \lambda_j \, u_j^H$$

where $u_j^H$ denotes the conjugate transpose of $u_j$. The computed eigenvectors are normalized to have Euclidean norm equal to $1$ and largest component real.
Balancing $A$ involves permuting its rows and columns to make it more nearly upper triangular and then scaling rows and columns by a diagonal similarity transformation to reduce the condition numbers of the eigenvalues and eigenvectors.
Computed reciprocal condition numbers pertain to the matrix after balancing. Permuting does not change condition numbers (in exact arithmetic), but scaling does.