Schur decomposition

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.

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 spectrum, 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 ⊂ ... ⊂ V_n = Cⁿ, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cⁿ) such that the first i basis vectors span V_i for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V₁,...,V_n).

A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue λ, corresponding to some eigenspace V_λ. Let V_λ^⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases Z₁ and Z₂ spanning V_λ and V_λ^⊥ respectively)

...
Wikipedia