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 .
References
- Nicholas J. Higham, Features of Matrices: Idea and Computation, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2008.
- C. F. Van Mortgage, A Research of the Matrix Exponential, Numerical Evaluation Report No. 10, College of Manchester, UK, 1975.