Schur decomposition

The complex Schur decomposition reads as follows: if is an 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 {{math|1={0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn}}, and that there exists an ordered orthonormal basis (for the standard Hermitian form of ) such that the first i basis vectors span 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 . There is also a real Schur decomposition. If is an square matrix with real entries, then A can be expressed as A = Q H Q^{-1} where is an orthogonal matrix and is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of and 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} 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) \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 Iλ is the identity operator on Vλ. 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 Vλ⊥, 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 λ be an eigenvalue of A, corresponding to some eigenspace Vλ. A induces an operator T on the quotient space Cn/Vλ. This operator is precisely the A22 submatrix from above. As before, T would have an eigenspace, say Wμ ⊂ Cn modulo Vλ. Notice the preimage of Wμ under the quotient map is an invariant subspace of A that contains Vλ. 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. == Notes ==

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. Its complexity is \mathcal{O}(n^3). == 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}. == References ==