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
![notag A [Q_1,~Q_2] = [Q_1,~Q_2] begin{bmatrix} T_{11} & T_{12} 0 & T_{22} end{bmatrix},](https://s0.wp.com/latex.php?latex=%5Cnotag+++A+%5BQ_1%2C%7EQ_2%5D+%3D+%5BQ_1%2C%7EQ_2%5D+++%5Cbegin%7Bbmatrix%7D+++T_%7B11%7D+%26+T_%7B12%7D+%5C%5C+++++++0++%26+T_%7B22%7D+%5C%5C+++%5Cend%7Bbmatrix%7D%2C+&bg=ffffff&fg=222222&s=0&c=20201002)
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
![R_{ii} = left[begin{array}{@{}rr@{mskip2mu}} a & b -b & a end{array}right], quad b ne 0,](https://s0.wp.com/latex.php?latex=R_%7Bii%7D+%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7B%40%7B%7Drr%40%7B%5Cmskip2mu%7D%7D+a+%26+b+%5C%5C++++++++++++++++-b+%26+a+%5Cend%7Barray%7D%5Cright%5D%2C+%5Cquad+b+%5Cne+0%2C+&bg=ffffff&fg=222222&s=0&c=20201002)
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.
