In linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.
Statement The complex Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as
A = Q U Q − 1 {\displaystyle A=QUQ^{-1}}
for some unitary matrix Q (so that the inverse Q−1 is also the conjugate transpose Q* of Q), and some upper triangular matrix U. This is called a Schur form of A. Since U is similar to A, it has the same spectrum, and since it is triangular, its 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 a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V1, ..., Vn). There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as A = Q H Q − 1 {\displaystyle A=QHQ^{-1}} where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is triangular. This is a stronger property than being Hessenberg. Just as in the complex case, a family of commuting real matrices {Ai} may be simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix Q such that, for every Ai in the given family,
H i = Q A i Q − 1 {\displaystyle H_{i}=QA_{i}Q^{-1}}
is upper quasi-triangular.
Proof 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 Vλ. Let Vλ⊥ 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 Vλ and Vλ⊥ respectively)
[ Z 1 Z 2 ] ∗ A [ Z 1 Z 2 ] = [ λ I λ A 12 0 A 22 ] : V λ ⊕ V λ ⊥ → V λ ⊕ V λ ⊥ {\displaystyle {\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}}
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