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 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 a n × n square matrix with complex entries, then A can be expressed as
where Q is a unitary matrix, 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, its eigenvalues are the diagonal entries of U.
The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces = V0 ⊂ V1 ⊂... ⊂ Vn = Cn, and that there exists an ordered orthonormal basis 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.

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
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 n times. 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.

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.
See the Nonsymmetric Eigenproblems section in LAPACK Users' Guide.

Applications

applications include:
Given square matrices A and B, the generalized Schur decomposition factorizes both matrices as and, 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 that solve the generalized eigenvalue problem 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 satisfies.