<s>
A	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
is	O
a	O
difference	O
equation	O
in	O
which	O
the	O
value	O
of	O
a	O
vector	O
(	O
or	O
sometimes	O
,	O
a	O
matrix	O
)	O
of	O
variables	O
at	O
one	O
point	O
in	O
time	O
is	O
related	O
to	O
its	O
own	O
value	O
at	O
one	O
or	O
more	O
previous	O
points	O
in	O
time	O
,	O
using	O
matrices	B-Architecture
.	O
</s>
<s>
is	O
an	O
example	O
of	O
a	O
second-order	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
,	O
in	O
which	O
is	O
an	O
vector	O
of	O
variables	O
and	O
and	O
are	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
most	O
commonly	O
encountered	O
matrix	B-Algorithm
difference	I-Algorithm
equations	I-Algorithm
are	O
first-order	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
where	O
it	O
is	O
assumed	O
that	O
is	O
invertible	O
.	O
</s>
<s>
The	O
first-order	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
is	O
stablethat	O
is	O
,	O
converges	O
asymptotically	O
to	O
the	O
steady	O
state	O
if	O
and	O
only	O
if	O
all	O
eigenvalues	O
of	O
the	O
transition	O
matrix	O
(	O
whether	O
real	O
or	O
complex	O
)	O
have	O
an	O
absolute	O
value	O
which	O
is	O
less	O
than	O
1	O
.	O
</s>
<s>
Then	O
we	O
can	O
iterate	O
and	O
substitute	O
repeatedly	O
from	O
the	O
initial	B-Algorithm
condition	I-Algorithm
,	O
which	O
is	O
the	O
initial	O
value	O
of	O
the	O
vector	O
and	O
which	O
must	O
be	O
known	O
in	O
order	O
to	O
find	O
the	O
solution	O
:	O
</s>
<s>
where	O
is	O
an	O
matrix	O
whose	O
columns	O
are	O
the	O
eigenvectors	O
of	O
(	O
assuming	O
the	O
eigenvalues	O
are	O
all	O
distinct	O
)	O
and	O
is	O
an	O
diagonal	B-Algorithm
matrix	I-Algorithm
whose	O
diagonal	O
elements	O
are	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
This	O
solution	O
motivates	O
the	O
above	O
stability	O
result	O
:	O
shrinks	O
to	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
over	O
time	O
if	O
and	O
only	O
if	O
the	O
eigenvalues	O
of	O
are	O
all	O
less	O
than	O
unity	O
in	O
absolute	O
value	O
.	O
</s>
<s>
Thus	O
each	O
individual	O
scalar	O
variable	O
of	O
an	O
-dimensional	O
first-order	O
linear	O
system	O
evolves	O
according	O
to	O
a	O
univariate	O
th-degree	O
difference	O
equation	O
,	O
which	O
has	O
the	O
same	O
stability	O
property	O
(	O
stable	O
or	O
unstable	O
)	O
as	O
does	O
the	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Matrix	B-Algorithm
difference	I-Algorithm
equations	I-Algorithm
of	O
higher	O
orderthat	O
is	O
,	O
with	O
a	O
time	O
lag	O
longer	O
than	O
one	O
periodcan	O
be	O
solved	O
,	O
and	O
their	O
stability	O
analyzed	O
,	O
by	O
converting	O
them	O
into	O
first-order	O
form	O
using	O
a	O
block	B-Algorithm
matrix	I-Algorithm
(	O
matrix	O
of	O
matrices	B-Architecture
)	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
is	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
linear-quadratic-Gaussian	O
control	O
,	O
there	O
arises	O
a	O
nonlinear	O
matrix	B-Architecture
equation	I-Architecture
for	O
the	O
reverse	O
evolution	O
of	O
a	O
current-and-future-cost	O
matrix	O
,	O
denoted	O
below	O
as	O
.	O
</s>
<s>
This	O
equation	O
is	O
called	O
a	O
discrete	O
dynamic	O
Riccati	O
equation	O
,	O
and	O
it	O
arises	O
when	O
a	O
variable	O
vector	O
evolving	O
according	O
to	O
a	O
linear	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
is	O
controlled	O
by	O
manipulating	O
an	O
exogenous	O
vector	O
in	O
order	O
to	O
optimize	O
a	O
quadratic	O
cost	O
function	O
.	O
</s>
<s>
The	O
parameter	O
matrices	B-Architecture
and	O
are	O
from	O
the	O
linear	O
equation	O
,	O
and	O
the	O
parameter	O
matrices	B-Architecture
and	O
are	O
from	O
the	O
quadratic	O
cost	O
function	O
.	O
</s>
<s>
In	O
most	O
contexts	O
the	O
evolution	O
of	O
backwards	O
through	O
time	O
is	O
stable	O
,	O
meaning	O
that	O
converges	O
to	O
a	O
particular	O
fixed	O
matrix	O
which	O
may	O
be	O
irrational	O
even	O
if	O
all	O
the	O
other	O
matrices	B-Architecture
are	O
rational	O
.	O
</s>
<s>
in	O
which	O
the	O
matrices	B-Architecture
are	O
all	O
.	O
</s>
<s>
so	O
by	B-Algorithm
induction	I-Algorithm
the	O
form	O
holds	O
for	O
all	O
.	O
</s>
