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Purpose


LA_GEES 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

\begin{displaymath}A = Z\,T\,Z^H.\end{displaymath}

Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of $Z$ then form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.
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

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

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.



next up previous contents index
Next: Arguments Up: Standard Nonsymmetric Eigenvalue Problems Previous: LA_GEES   Contents   Index
Susan Blackford 2001-08-19