Schur decomposition

In the mathematical discipline of 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 equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.

Statement
The Schur decomposition reads as follows: if $A$ is an $n × n$ square matrix with complex entries, then A can be expressed as $$ A = Q U Q^{-1}$$ 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 ${0} = V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{n} = C^{n}$, and that there exists an ordered orthonormal basis (for the standard Hermitian form of $C^{n}$) 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 $(V_{1}, ..., V_{n})$.

Proof
A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue &lambda;, corresponding to some eigenspace V&lambda;. Let V&lambda;⊥ 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&lambda; and V&lambda;⊥ respectively) $$\begin{bmatrix} Z_1 & Z_2 \end{bmatrix}^{*} A \begin{bmatrix}Z_1 & Z_2\end{bmatrix} = \begin{bmatrix} \lambda \, I_{\lambda} & A_{12} \\ 0 & A_{22} \end{bmatrix}: \begin{matrix} V_{\lambda} \\ \oplus \\ V_{\lambda}^{\perp} \end{matrix} \rightarrow \begin{matrix} V_{\lambda} \\ \oplus \\ V_{\lambda}^{\perp} \end{matrix} $$ where I&lambda; is the identity operator on V&lambda;. The above matrix would be upper-triangular except for the A22 block. But exactly the same procedure can be applied to the sub-matrix A22, viewed as an operator on V&lambda;⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most n steps. Thus the space Cn will be exhausted and the procedure has yielded the desired result.

The above argument can be slightly restated as follows: let &lambda; be an eigenvalue of A, corresponding to some eigenspace V&lambda;. A induces an operator T on the quotient space Cn/V&lambda;. This operator is precisely the A22 submatrix from above. As before, T would have an eigenspace, say W&mu; ⊂ Cn modulo V&lambda;. Notice the preimage of W&mu; under the quotient map is an invariant subspace of A that contains V&lambda;. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that A stabilizes.

Computation
The Schur decomposition of a given matrix is numerically computed by the QR algorithm or its variants. In other words, the roots of the characteristic polynomial corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the QR algorithm can be used to compute the roots of any given characteristic polynomial by finding the Schur decomposition of its companion matrix. Similarly, the QR algorithm is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the QR algorithm is formally an infinite sequence of operations, convergence to machine precision is practically achieved in $\mathcal{O}(n^3)$ operations. See the Nonsymmetric Eigenproblems section in LAPACK Users' Guide.

Applications
Lie theory applications include:
 * Every invertible operator is contained in a Borel group.
 * Every operator fixes a point of the flag manifold.

Generalized Schur decomposition
Given square matrices A and B, the generalized Schur decomposition factorizes both matrices as $$A = QSZ^*$$ and $$B = QTZ^*$$, where Q and Z are unitary, and S and T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition.

The generalized eigenvalues $$\lambda$$ that solve the generalized eigenvalue problem $$A\mathbf{x}=\lambda B\mathbf{x}$$ (where x is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of S to those of T. That is, using subscripts to denote matrix elements, the ith generalized eigenvalue $$\lambda_i$$ satisfies $$\lambda_i = S_{ii} / T_{ii}$$.