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Schur Factorization
Syntax B = schur(X)
[UB] = schur(X)
See Also eigen , svd , qred , lu

Description
Computes a schur factorization of the matrix square matrix x, where X is real, double-precision, or complex. A schur factorization of X is a pair of matrices U and B such that
     X = U B conj(U)'
where U is a unitary matrix , the matrix B is in Schur form, and conj(U)' denotes the complex conjugate transpose of U.

The return values B and U have the same type and dimension as the matrix X. Note that because U is a unitary matrix, the matrices X and B have the same eigenvalues.

Real Schur Form
A real or double-precision matrix is in schur form if it is block diagonal with each diagonal block either a x 1 or x 2 matrix. In addition each of the x 2 blocks has the form
     / a  b \
     \ c  a /
where b c < 0. The eigenvalues corresponding to such a block is
        +   -----
     a  - \/ b c
The eigenvalues corresponding to a x 1 blocks is the value on the diagonal.

Complex Schur Form
A complex matrix is in schur form if it is upper triangular with its eigenvalues on the diagonal of the matrix.

Example
You can compute a schur factorization of the matrix
     / 2 1 \
     \ 1 2 /
by entering
     X = {[2., 1.], [1., 2.]}
     [U, B] = schur(X)
     print B
which returns
     {
     [ 3 , 0 ]
     [ 0 , 1 ]
     }
If you continue this example by entering
     U * B * U'
O-Matrix will respond
     {
     [ 2 , 1 ]
     [ 1 , 2 ]
     }
(Note that because X is not complex, U is not complex and it is not necessary to take the complex conjugate of U in the example above.)