A Schur decomposition of a matrix is a factorization , the place is unitary and is higher triangular. The diagonal entries of are the eigenvalues of , and they are often made to look in any order by selecting appropriately. The columns of are known as Schur vectors.
A subspace of is an invariant subspace of if for all . If we partition and conformably we are able to write
which provides , exhibiting that the columns of span an invariant subspace of . Moreover, . The primary column of is an eigenvector of comparable to the eigenvalue , however the different columns usually are not eigenvectors, normally. Eigenvectors will be computed by fixing higher triangular methods involving , the place is an eigenvalue.
Write , the place and is strictly higher triangular. Taking Frobenius norms provides , or
Therefore is impartial of the actual Schur decomposition and it supplies a measure of the departure from normality. The matrix is regular (that’s, ) if and provided that . So a standard matrix is unitarily diagonalizable: .
An necessary utility of the Schur decomposition is to compute matrix features. The relation reveals that computing reduces to computing a perform of a triangular matrix. Matrix features illustrate what Van Mortgage (1975) describes as “probably the most fundamental tenets of numerical algebra”, particularly “something that the Jordan decomposition can do, the Schur decomposition can do higher!”. Certainly the Jordan canonical kind is constructed on a probably sick conditioned similarity transformation whereas the Schur decomposition employs a superbly conditioned unitary similarity, and the complete higher triangular issue of the Schur kind can do most of what the Jordan kind’s bidiagonal issue can do.
An higher quasi-triangular matrix is a block higher triangular matrix
whose diagonal blocks are both or . An actual matrix has a actual Schur decomposition through which through which all of the elements are actual, is orthogonal, and is higher quasi-triangular with any diagonal blocks having advanced conjugate eigenvalues. If is regular then the blocks have the shape
which has eigenvalues .
The Schur decomposition will be computed by the QR algorithm at a value of about flops for and , or flops for solely.
In MATLAB, the Schur decomposition is computed with the
schur perform: the command
[Q,T] = schur(A) returns the true Schur kind if is actual and in any other case the advanced Schur kind. The advanced Schur kind for an actual matrix will be obtained with
[Q,T] = schur(A,'advanced'). The
rsf2csf perform converts the true Schur kind to the advanced Schur kind. The=
ordschur perform takes a Schur decomposition and modifies it in order that the eigenvalues seem in a specified order alongside the diagonal of .