<s>
In	O
mathematics	O
,	O
in	O
the	O
field	O
of	O
control	O
theory	O
,	O
a	O
Sylvester	B-Algorithm
equation	I-Algorithm
is	O
a	O
matrix	B-Architecture
equation	I-Architecture
of	O
the	O
form	O
:	O
</s>
<s>
A	O
Sylvester	B-Algorithm
equation	I-Algorithm
has	O
a	O
unique	O
solution	O
for	O
X	O
exactly	O
when	O
there	O
are	O
no	O
common	O
eigenvalues	O
of	O
A	O
and−B	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
condition	O
for	O
the	O
uniqueness	O
of	O
a	O
solution	O
X	O
is	O
almost	O
the	O
same	O
:	O
There	O
exists	O
a	O
unique	O
solution	O
X	O
exactly	O
when	O
the	O
spectra	O
of	O
A	O
and	O
−B	O
are	O
disjoint	B-Algorithm
.	O
</s>
<s>
where	O
is	O
of	O
dimension	O
,	O
is	O
of	O
dimension	O
,	O
of	O
dimension	O
and	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Given	O
matrices	O
and	O
,	O
the	O
Sylvester	B-Algorithm
equation	I-Algorithm
has	O
a	O
unique	O
solution	O
for	O
any	O
if	O
and	O
only	O
if	O
and	O
do	O
not	O
share	O
any	O
eigenvalue	O
.	O
</s>
<s>
where	O
denotes	O
the	O
spectrum	O
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
theorem	O
remains	O
true	O
for	O
real	B-Architecture
matrices	I-Architecture
with	O
the	O
caveat	O
that	O
one	O
considers	O
their	O
complex	O
eigenvalues	O
.	O
</s>
<s>
Given	O
two	O
square	O
complex	O
matrices	O
A	O
and	O
B	O
,	O
of	O
size	O
n	O
and	O
m	O
,	O
and	O
a	O
matrix	B-Architecture
C	O
of	O
size	O
n	O
by	O
m	O
,	O
then	O
one	O
can	O
ask	O
when	O
the	O
following	O
two	O
square	O
matrices	O
of	O
size	O
n+m	O
are	O
similar	B-Algorithm
to	O
each	O
other	O
:	O
and	O
.	O
</s>
<s>
The	O
answer	O
is	O
that	O
these	O
two	O
matrices	O
are	O
similar	B-Algorithm
exactly	O
when	O
there	O
exists	O
a	O
matrix	B-Architecture
X	O
such	O
that	O
AX−XB	O
=	O
C	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
X	O
is	O
a	O
solution	O
to	O
a	O
Sylvester	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
This	O
is	O
known	O
as	O
Roth	B-Algorithm
's	I-Algorithm
removal	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Roth	B-Algorithm
's	I-Algorithm
removal	I-Algorithm
rule	I-Algorithm
does	O
not	O
generalize	O
to	O
infinite-dimensional	O
bounded	O
operators	O
on	O
a	O
Banach	O
space	O
.	O
</s>
<s>
A	O
classical	O
algorithm	O
for	O
the	O
numerical	O
solution	O
of	O
the	O
Sylvester	B-Algorithm
equation	I-Algorithm
is	O
the	O
Bartels	B-Algorithm
–	I-Algorithm
Stewart	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
consists	O
of	O
transforming	O
and	O
into	O
Schur	O
form	O
by	O
a	O
QR	O
algorithm	O
,	O
and	O
then	O
solving	O
the	O
resulting	O
triangular	O
system	O
via	O
back-substitution	B-Algorithm
.	O
</s>
<s>
This	O
algorithm	O
,	O
whose	O
computational	O
cost	O
is	O
arithmetical	O
operations	O
,	O
is	O
used	O
,	O
among	O
others	O
,	O
by	O
LAPACK	B-Application
and	O
the	O
lyap	O
function	O
in	O
GNU	B-Language
Octave	I-Language
.	O
</s>
<s>
In	O
some	O
specific	O
image	O
processing	O
application	O
,	O
the	O
derived	O
Sylvester	B-Algorithm
equation	I-Algorithm
has	O
a	O
closed	O
form	O
solution	O
.	O
</s>
