"why Schur vectors": stability + they're always computed when you compute eigenvectors. For non-symmetric matrices, the standard approach to compute an eigen d

"why Schur vectors": stability + they're always computed when you compute eigenvectors. For non-symmetric matrices, the standard approach to compute an eigen decomposition is the QR algorithm, which is an iterative method that converges to a Schur decomposition AQ=QR. So, it's always cheaper to compute only Schur vectors. Eigenvectors are obtained by computing the eigen decomposition RY = YΛ (easy for R diagonal): A(QY) = (QY)Λ is the eigen decomposition of A. The QR algorithm works well because it uses stable operations throughout: householder reflections and given's rotations. Those preserve norm, so they don't amplify rounding errors. The Schur decomposition itself is useful for the same reason: Schur vectors are an orthonormal basis for eigenvectors. If you need to project something onto eigenvectors, it's more stable to work with Schur vectors than eigenvectors. The last step RY = YΛ is done implicitly, with a backwards solve involving R. If that matrix is badly conditioned, the eigenvectors have large errors.