Schur decomposition

Question

If $A$ is real and nonsymmetric with Schur decomposition $UTU^H$, then what types of matrices are $U$ and $T$? How are the eigenvalues of $A$ related to $U$ and $T$?

Answer 1

You have two possibilities:

1. (Complex) Schur decomposition: $U$ and $T$ are complex, with $U$ being unitary ($U^*U = {\rm I}$) and $T$ being (usually upper) triangular.
2. Real Schur decomposition: $U$ and $T$ are real, with $U$ being orthogonal ($U^TU = {\rm I}$) and $T$ being (usually upper) quasitriangular (it has diagonal blocks of order $1$ and $2$).

In both cases, since the Schur decomposition is a (unitary/orthogonal) similarity of $A$ and $T$, these two have the same eigenvalues.

In case 1 the diagonal elements of $T$ are the eigenvalues of $T$ (and, consequently, of $A$). In case 2, the eigenvalues are the elements of the diagonal blocks of $T$ of order $1$ and the eigenvalues of the diagonal blocks of $T$ of order $2$ (these eigenvalues are complex conjugate pairs and can be computed directly).

Given a normal matrix $A$, columns of $U$ in case 1 are its eigenvectors.

If $A$ is symmetric, then it has only real eigenvalues, so cases 1 and 2 are the same: $U$ is real unitary (meaning orthogonal), $T$ is real diagonal (with the eigenvalues on the diagonal), and the columns of $U$ are (orthonormal) eigenvectors of $A$.