<s>
In	O
mathematics	O
,	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
used	O
to	O
describe	O
the	O
transitions	O
of	O
a	O
Markov	O
chain	O
.	O
</s>
<s>
It	O
is	O
also	O
called	O
a	O
probability	B-Algorithm
matrix	I-Algorithm
,	O
transition	O
matrix	O
,	O
substitution	B-Algorithm
matrix	I-Algorithm
,	O
or	O
Markov	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
stochastic	B-Algorithm
matrix	I-Algorithm
was	O
first	O
developed	O
by	O
Andrey	O
Markov	O
at	O
the	O
beginning	O
of	O
the	O
20th	O
century	O
,	O
and	O
has	O
found	O
use	O
throughout	O
a	O
wide	O
variety	O
of	O
scientific	O
fields	O
,	O
including	O
probability	O
theory	O
,	O
statistics	O
,	O
mathematical	O
finance	O
and	O
linear	B-Language
algebra	I-Language
,	O
as	O
well	O
as	O
computer	B-General_Concept
science	I-General_Concept
and	O
population	O
genetics	O
.	O
</s>
<s>
There	O
are	O
several	O
different	O
definitions	O
and	O
types	O
of	O
stochastic	B-Algorithm
matrices	I-Algorithm
:	O
</s>
<s>
A	O
right	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
a	O
real	B-Algorithm
square	I-Algorithm
matrix	I-Algorithm
,	O
with	O
each	O
row	O
summing	O
to	O
1	O
.	O
</s>
<s>
A	O
left	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
a	O
real	B-Algorithm
square	I-Algorithm
matrix	I-Algorithm
,	O
with	O
each	O
column	O
summing	O
to	O
1	O
.	O
</s>
<s>
A	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
nonnegative	O
real	O
numbers	O
with	O
each	O
row	O
and	O
column	O
summing	O
to	O
1	O
.	O
</s>
<s>
Thus	O
,	O
each	O
row	O
of	O
a	O
right	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
(	O
or	O
column	O
of	O
a	O
left	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
)	O
is	O
a	O
stochastic	O
vector	O
.	O
</s>
<s>
A	O
common	O
convention	O
in	O
English	O
language	O
mathematics	O
literature	O
is	O
to	O
use	O
row	O
vectors	O
of	O
probabilities	O
and	O
right	O
stochastic	B-Algorithm
matrices	I-Algorithm
rather	O
than	O
column	O
vectors	O
of	O
probabilities	O
and	O
left	O
stochastic	B-Algorithm
matrices	I-Algorithm
;	O
this	O
article	O
follows	O
that	O
convention	O
.	O
</s>
<s>
The	O
stochastic	B-Algorithm
matrix	I-Algorithm
was	O
developed	O
alongside	O
the	O
Markov	O
chain	O
by	O
Andrey	O
Markov	O
,	O
a	O
Russian	O
mathematician	O
and	O
professor	O
at	O
St.	O
Petersburg	O
University	O
who	O
first	O
published	O
on	O
the	O
topic	O
in	O
1906	O
.	O
</s>
<s>
Stochastic	B-Algorithm
matrices	I-Algorithm
were	O
further	O
developed	O
by	O
scholars	O
like	O
Andrey	O
Kolmogorov	O
,	O
who	O
expanded	O
their	O
possibilities	O
by	O
allowing	O
for	O
continuous-time	O
Markov	O
processes	O
.	O
</s>
<s>
By	O
the	O
1950s	O
,	O
articles	O
using	O
stochastic	B-Algorithm
matrices	I-Algorithm
had	O
appeared	O
in	O
the	O
fields	O
of	O
econometrics	O
and	O
circuit	O
theory	O
.	O
</s>
<s>
In	O
the	O
1960s	O
,	O
stochastic	B-Algorithm
matrices	I-Algorithm
appeared	O
in	O
an	O
even	O
wider	O
variety	O
of	O
scientific	O
works	O
,	O
from	O
behavioral	O
science	O
to	O
geology	O
to	O
residential	O
planning	O
.	O
</s>
<s>
In	O
addition	O
,	O
much	O
mathematical	O
work	O
was	O
also	O
done	O
through	O
these	O
decades	O
to	O
improve	O
the	O
range	O
of	O
uses	O
and	O
functionality	O
of	O
the	O
stochastic	B-Algorithm
matrix	I-Algorithm
and	O
Markovian	O
processes	O
more	O
generally	O
.	O
</s>
<s>
From	O
the	O
1970s	O
to	O
present	O
,	O
stochastic	B-Algorithm
matrices	I-Algorithm
have	O
found	O
use	O
in	O
almost	O
every	O
field	O
that	O
requires	O
formal	O
analysis	O
,	O
from	O
structural	O
science	O
to	O
medical	O
diagnosis	O
to	O
personnel	O
management	O
.	O
</s>
<s>
In	O
addition	O
,	O
stochastic	B-Algorithm
matrices	I-Algorithm
have	O
found	O
wide	O
use	O
in	O
land	O
change	O
modeling	O
,	O
usually	O
under	O
the	O
term	O
Markov	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
stochastic	B-Algorithm
matrix	I-Algorithm
describes	O
a	O
Markov	O
chain	O
over	O
a	O
finite	O
state	O
space	O
with	O
cardinality	B-Application
.	O
</s>
<s>
If	O
the	O
probability	O
of	O
moving	O
from	O
to	O
in	O
one	O
time	O
step	O
is	O
,	O
the	O
stochastic	B-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
using	O
as	O
the	O
-th	O
row	O
and	O
-th	O
column	O
element	O
,	O
e.g.	O
,	O
</s>
<s>
thus	O
this	O
matrix	O
is	O
a	O
right	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Using	O
this	O
,	O
it	O
can	O
be	O
seen	O
that	O
the	O
product	O
of	O
two	O
right	O
stochastic	B-Algorithm
matrices	I-Algorithm
and	O
is	O
also	O
right	O
stochastic	O
:	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
-th	O
power	O
of	O
a	O
right	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
is	O
also	O
right	O
stochastic	O
.	O
</s>
<s>
A	O
stationary	O
probability	O
vector	O
is	O
defined	O
as	O
a	O
distribution	O
,	O
written	O
as	O
a	O
row	O
vector	O
,	O
that	O
does	O
not	O
change	O
under	O
application	O
of	O
the	O
transition	O
matrix	O
;	O
that	O
is	O
,	O
it	O
is	O
defined	O
as	O
a	O
probability	O
distribution	O
on	O
the	O
set	O
which	O
is	O
also	O
a	O
row	O
eigenvector	O
of	O
the	O
probability	B-Algorithm
matrix	I-Algorithm
,	O
associated	O
with	O
eigenvalue	O
1	O
:	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
spectral	O
radius	O
of	O
any	O
stochastic	B-Algorithm
matrix	I-Algorithm
is	O
one	O
.	O
</s>
<s>
By	O
the	O
Gershgorin	O
circle	O
theorem	O
,	O
all	O
of	O
the	O
eigenvalues	O
of	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
have	O
absolute	O
values	O
less	O
than	O
or	O
equal	O
to	O
one	O
.	O
</s>
<s>
Additionally	O
,	O
every	O
right	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
has	O
an	O
"	O
obvious	O
"	O
column	O
eigenvector	O
associated	O
to	O
the	O
eigenvalue	O
1	O
:	O
the	O
vector	O
,	O
whose	O
coordinates	O
are	O
all	O
equal	O
to	O
1	O
(	O
just	O
observe	O
that	O
multiplying	O
a	O
row	O
of	O
times	O
equals	O
the	O
sum	O
of	O
the	O
entries	O
of	O
the	O
row	O
and	O
,	O
hence	O
,	O
it	O
equals	O
1	O
)	O
.	O
</s>
<s>
As	O
left	O
and	O
right	O
eigenvalues	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
are	O
the	O
same	O
,	O
every	O
stochastic	B-Algorithm
matrix	I-Algorithm
has	O
,	O
at	O
least	O
,	O
a	O
row	O
eigenvector	O
associated	O
to	O
the	O
eigenvalue	O
1	O
and	O
the	O
largest	O
absolute	O
value	O
of	O
all	O
its	O
eigenvalues	O
is	O
also	O
1	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
Perron	O
–	O
Frobenius	O
theorem	O
also	O
ensures	O
that	O
every	O
irreducible	O
stochastic	B-Algorithm
matrix	I-Algorithm
has	O
such	O
a	O
stationary	O
vector	O
,	O
and	O
that	O
the	O
largest	O
absolute	O
value	O
of	O
an	O
eigenvalue	O
is	O
always	O
1	O
.	O
</s>
<s>
However	O
,	O
for	O
a	O
matrix	O
with	O
strictly	O
positive	O
entries	O
(	O
or	O
,	O
more	O
generally	O
,	O
for	O
an	O
irreducible	O
aperiodic	O
stochastic	B-Algorithm
matrix	I-Algorithm
)	O
,	O
this	O
vector	O
is	O
unique	O
and	O
can	O
be	O
computed	O
by	O
observing	O
that	O
for	O
any	O
we	O
have	O
the	O
following	O
limit	O
,	O
</s>
<s>
Intuitively	O
,	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
represents	O
a	O
Markov	O
chain	O
;	O
the	O
application	O
of	O
the	O
stochastic	B-Algorithm
matrix	I-Algorithm
to	O
a	O
probability	O
distribution	O
redistributes	O
the	O
probability	O
mass	O
of	O
the	O
original	O
distribution	O
while	O
preserving	O
its	O
total	O
mass	O
.	O
</s>
<s>
We	O
use	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
,	O
(	O
below	O
)	O
,	O
to	O
represent	O
the	O
transition	O
probabilities	O
of	O
this	O
system	O
(	O
rows	O
and	O
columns	O
in	O
this	O
matrix	O
are	O
indexed	O
by	O
the	O
possible	O
states	O
listed	O
above	O
,	O
with	O
the	O
pre-transition	O
state	O
as	O
the	O
row	O
and	O
post-transition	O
state	O
as	O
the	O
column	O
)	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
represents	O
a	O
column	O
matrix	O
of	O
all	O
ones	O
that	O
acts	O
as	O
a	O
sum	O
over	O
states	O
.	O
</s>
