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Syntax |
B = schur( X) |
[ U, B] = schur( X) | |
See Also | eigen , svd , qred , lu |
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.
1 x 1
or
2 x 2
matrix.
In addition each of the 2 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 1 x 1
blocks is the value on the diagonal.
/ 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.)