Purpose

LA_GEESX computes for a real/complex square matrix $A$, the eigenvalues, the real-Schur/complex-Schur form $T$, and, optionally, the matrix of Schur vectors $Z$, where $Z$ is orthogonal/unitary. This gives the Schur factorization

$$A = Z\,T\,Z^H.$$

Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left, computes a reciprocal condition number for the average of the selected eigenvalues, and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues. The leading columns of $Z$ form an orthonormal basis for this invariant subspace.
A real matrix is in real-Schur form if it is block upper triangular with $1 \times 1$ and $2 \times 2$ blocks along the main diagonal. $2 \times 2$ blocks are standardized in the form

$$\left[ \begin{array}{cc} a & b \\ c & a \end{array} \right]$$

where $b \, c < 0$. The eigenvalues of such a block are $a \pm \sqrt{bc}$.
A complex matrix is in complex-Schur form if it is upper triangular.