Statement
The Schur decomposition reads as follows: if A is a n × n square matrix with complex entries, then A can be expressed as
where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A. Since U is similar to A, it has the same multiset of eigenvalues, and since it is triangular, those eigenvalues are the diagonal entries of U.
The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ... ⊂ Vn = Cn, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cn) such that the first i basis vectors span Vi for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that an operator T on a complex finite-dimensional vector space stabilizes a complete flag (V1,...,Vn).
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