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Stochastic Matrices in a Finite Field
Abstract: In this project we will explore the properties of stochastic matrices in
both the real and the finite fields. We first explore what properties 2  2 stochastic
matrices in the real field have and then exam if they hold in the finite fields. We will
prove how, given the conditions of a finite field, properties hold or fail to hold. We will
extend our scope to 3  3 stochastic matrices and lay the groundwork for future
research. Finally, we show how the properties in the real field extend directly to another
infinite field: the rationals.
Introduction
Stochastic processes can be used to model many phenomena involving events
(states) that evolve over time. This phenomena can be the behavior of the price of a
stock given a set of time periods or weather patterns. In precise terms, a stochastic
process is an, “indexed collection of random variables (Hillier)” where the index is
usually the nonnegative integers. We narrow our focus to stochastic matrices, which are
particularly useful in defining transition probabilities. Knowing the properties of such
matrices allows to define solutions to problems that involve their use, among other
things. Its application aside, however, we shall study their behavior in finite fields.
Literature review and Properties of Stochastic
Matrices in 
A stochastic matrix is a square matrix such that for all of its
n
entries, p ij , 0 ≤ p ij ≤ 1, and that ∑ p ij  1, for all j (a column stochastic matrix),
i1
n
or
n
n
ji
i1
∑ p ij  1 for all i ( a row stochastic matrix), or ∑ p ij  1 and ∑ p ij  1 for all i
j1
and j, respectively (a doubly stochastic matrix).
Property 1
Consider a row stochastic n  n matrix, P, and its eigenvalues. It is clear that Pe  e
where e 
T
. Hence, 1, e is always an eigenpair of a row stochastic
1  1
matrix. In the case of the column stochastic matrix, say, P T , we similarly have
e T P T  e T . For a doubly stochastic matrix P, both Pe  e and e T P T  e T hold.
Property 2
Consider the 1-norm and the -norm of the three types of stochastic matrices.
Recall that the 1-norm of an n  n matrix, A  a ij , is defined as ||A|| 1  max 1≤j≤n 
1
n
∑
n
| a ij | and its -norm is defined as ||A||   max 1≤i≤n  ∑ | a ij |. Simply, put, the
i1
i1
former is the maximum column sum and the latter is the maximum row sum.
The -norm for a row stochastic matrix P is hence equal to one and its
1-norm, ||P|| 1 : 1 ≤ ||P|| 1 ≤ n. It is clear why for the case of the 1-norm the value is less
than n. To see why the lower limit is 1, consider the fact that the sum of all the entries
is n so if all column sums are less than 1, then the sum of all the column sums is less
than n: a contradiction. For a column stochastic matrix P, since P T is a row stochastic
matrix, we have ||P|| 1  1 and 1 ≤ ‖P‖  ≤ n. Consequently, for a doubly stochastic
matrix, P, ‖P‖ 1  ‖P‖   1.
Property 3
The spectral radius of an n  n matrix A is defined as A  max i1,...,n | i |
where  i for i  1, 2. . . n are eigenvalues of A. We know that 1, e will always exist as
an eigenpair for any stochastic matrix. Now, the absolute value of any eigenvalue
cannot be greater than any norm of the matrix to which it is an eigenvalue. Hence, the
spectral radius of any stochastic matrix is 1 since we know that 1, e is an eigenpair of
any stochastic matrix and any stochastic matrix either has an  −norm or 1-norm that
equals 1.
Property 4
It is known that for any square matrix, the sum of our stochastic matrices’
eigenvalues equal its trace and the product of their eigenvalues equal their
determinants. We prove these two results for 2  2 matrices as follows.
a b
Let A 
. Then the characteristic polynomial of A is
c d
detA − I  a − d −  − bc   2 − a  d  ad − bc  0. (1)
Let  1 and  2 be eigenvalues of A. Because  1 and  2 are roots of detA − I  0,
detA − I   −  1  −  2    2 −  1   2    1  2  0. (2)
Comparing coefficients of  and constant terms in (1) and (2), we obtain
 1   2  a  d  traceA, and  1  2  ad − bc  detA.
Property 5
For a 2  2 row stochastic matrix, if all of its entries are strictly greater than 0 then
eigenvalue  1  1 is simple.
a
1−a
where
Let P be a 2  2 row stochastic matrix. Then P 
1−b
b
0  a, b  1. Let  2 be another eigenvalue of P. Because  1   2  a  b,
 2  a  b − 1. Since 0  a, b  1, a  b − 1  2 − 1  1.
2
Property 6
Observe from Property 5 that  2  0 if and only if a  b  1, when
a b
. Hence, for a 2  2 row stochastic matrix whose rows are not linearly
a b
independent,  2  0. Let P be a 2  2 doubly stochastic matrix. Then
a
1−a
and  2  2a − 1.  1  1 is a simple eigenvalue if and only if
P
1−a
a
2a − 1  1 or a  1.
P
Empirical Work:
All of these properties rely on a crucial assumption: all the entries are real numbers.
Within the scope of elementary mathematical methods, we shall now explore the
properties of stochastic matrices with entries in finite fields. We first restrict our
analysis to 2  2 matrices with entries in ℤ p , where p is a prime number. We will also
establish that property 1 holds in any finite field ℤ p as well.
1. Row Stochastic Matrices in ℤ 2 :
There are only 2  2  2  2  16 possible matrices of which eight are either row,
column or doubly stochastic. we shall restrict our attention to row stochastic matrices,
inevitably a superset of doubly stochastic matrices.
Matrices
1 0
1 0
0 1
0 1
, P2 
, P3 
and P 4 
are all such
P1 
0 1
1 0
0 1
1 0
row stochastic matrices. Because the first three matrices are either lower or upper
triangular matrices, their eigenvalues are their diagonal elements which are 1 or 0. Their
eigenvectors can be obtained by solving the system of linear equations:
A − IX  0. Hence, eigenpairs of P 1 , P 2 and P 3 are:
1
0
, 1,
;
P 1 : 1,
1
1
P2 :
1,
P3 :
1,
1
1
1
1
,
0,
,
0,
0
1
1
0
; and
.
Eigenvalue  2 of P 4 satisfies the equation:  2  0 − 1  −1 ≡ 2 1. Let v 
x
y
be an eigenvector corresponding to eigenvalue 1. Then the equation P 4 v  v implies that
3
P4v 
eigenpair:
y
x

x
, that is, x  y. Hence, P 4 has only one
1
.
1
Clearly, Property 1: 1, e is an eigenpair of P, and Property 3: P  1, hold for
row stochastic matrices in ℤ 2 . Since in ℤ 2 , there is no row stochastic matrix with all
non-zero entries, Property 5 cannot be applied.
2. Row Stochastic Matrices in ℤ 3 :
In ℤ 3 , we have a total of 81 matrices to begin with. Of the matrices, we will again
consider only the row stochastic matrices. The rows a, b and c, d of such matrices
should hence be one of the pairs of values 1, 0 or 2, 2 and we have five such matrices,
which include P i , i  1, 2, 3, 4 that we analyzed in ℤ 2 . The additional five row stochastic
matrices are:
P5 
1,
y
1 0
2 2
, P6 
0 1
2 2
, P7 
2 2
1 0
, P8 
2 2
0 1
2 2
, P9 
2 2
Eigenvalues of P 5 and P 8 (they are lower and upper triangular matrices, respectively) are
0 1
. In
1 and 2. So, Property 3 no longer holds. Let us revisit P 4 
1 0
ℤ 3 ,  2  0 − 1  −1 ≡ 3 2. P 4 also has eigenvalues 1 and 2.
Consider the stochastic matrix whose entries are all strictly nonzero in ℤ 3 :
2 2
. In this case,  2  4 − 1  3 ≡ 3 0. So, both Property 3 and Property
P9 
2 2
5 hold.
3. Row Stochastic Matrices in ℤ 5 :
For ℤ 5 , we will deal with 525 possible matrices. Clearly, the row stochastic matrices
must have in their row entries a, b, c, d should be one of the pairs
3, 3, 4, 2, or 1, 0. Through simple computation, we have 25 row stochastic matrices.
Rather than calculating the eigenpairs of every such matrix, to verify whether the
properties hold for ℤ 5 , we shall first try to find simple counterexamples. In ℤ 5 , we can
4 2
for  2  4  4 − 1  7 ≡ 5 2.
indeed find a counterexample to Property 3:
2 4
3 3
for  2  3  4 − 1  6 ≡ 5 1.
2 4
We can therefore conclude that while Property 1 holds true in any finite field ℤ p ,
Properties 3 and Property 4 do not always hold. In the following, we shall analyze why
indeed certain properties hold.
The counterexample to Property 5:
General Results:
4
.
Row stochastic matrices in the finite fieldZ p where p is a prime number have an
eigenvalue of 0 if and only if det
a 1−a
b 1−b
trivial, the only matrix being of the form
 0. While the case in the real field is
a 1−a
a 1−a
, this will hold true in ℤ p as
long as the rows are linearly dependent.
Let a, 1 − a and b, 1 − b ∈ Z p ⊕ Z p such that there exists a k ≥ 1 where
a, 1 − a k  b, 1 − b so ak  band k1 − a  1 − b. Then
a 1−a
 a1 − b − b1 − a  a − ab − b  ba  a − b so a  b and like
det
b 1−b
a 1−a
in the real case we have the same form as we do in the real case:
.
a 1−a
Notice that all the eigenvalues obtained in the examples above have solutions in Z p .
We will prove that all row stochastic matrices have such a property. Given the row
a
1−a
, our characteristic equation will be
stochastic matrix P 
1−b
b
a − b −  − 1 − a1 − b  0. We know that   1, is a solution and it is clear
that  2  a  b − 1 ∈ Z p . Hence all eigenvalues for 2  2 row stochastic matrices exist
within the finite field to which the stochastic matrix is restricted. We can see that
property 6 holds here as well: 1  a  b − 1  a  b  trP and
a  b − 1  detP  ab − 1 − a1 − b  ab − 1 − a − b  ab  a  b  1.
Doubly stochastic matrices will also have the form
a
1−a
, which
1−a
a
will yield the characteristic equation a −  2 − 1 − a 2  0 and here, as it held in the
real case,  1,2  1, 2a − 1 and, again,   1 is simple if and only if a  0.
Unlike stochastic matrices in the real field, it is possible to have an eigenvalue
greater than 1 in the finite field. We know this from one of the examples above. We
also know the solutions to the characteristic equations of any stochastic matrix belong
to the finite field and hence, no field extensions are necessary to have solutions.
3  3 Stochastic Matrices in ℤ 2 and ℤ 3
We can extend our scope to 3  3 stochastic matrices and see how the properties we
have derived applies to them. It is still obvious that 1, e is an eigenpair to this
5
matrix:
a b c
1
d e f
1
1

1
since a  b  c  d  e  f  g  h  i  1.
g h i
1
1
Considering row stochastic matrices in ℤ 2 , the only possible row entries we have are
1, 0, 0 and 1, 1, 1. For ℤ 3 , we have 2, 1, 1, 2, 2, 0, 1, 0, 0. Let us look into their
eigenvalues. We know that if the solutions to their characteristic equations are all
within their fields (without extensions), we will have a characteristic equations of the
a b c
form  − 1 −  2  −  3 . We know that if
is linearly dependent,
d e f
g h i
a b c
det
d e f
 0 and like in the 2  2 case, we have an eigenvalue equal to 0 and
g h i
the other must be in ℤ 3 . While it would be more challenging to find the general form in
which 3  3 matrices would have simple or distinct eigenvalues we can still apply some
of the properties from above. For instance, the 3  3 identity matrix clearly has a
nonzero determinant and its characteristic equation is ( − 1 3  0. This holds true for
both ℤ 2 and ℤ 3 . Linear independence is hence a necessary but not sufficient condition
for the eigenvalue of 1 to be simple.
Stochastic Matrices in the Field of Rational Numbers
It is obvious that Properties 1, 2, 3 hold in the field of rationals. Indeed all
properties that held in the real field hold also in the field of rationals as well. However,
we shall explore their characteristic equation and solutions and find that they take on
well-defined forms.
a
1−a
p
where a  q and b  rs , and
Consider row stochastic matrix
1−b
b
p, q, r and s are integers. Then
p
sp  rq − qs
 2  a  b − 1  q  rs − 1 
qs
is again a rational number. For a doubly stochastic matrix
a
1−a
1−a
a
where
p
a  q , and p, q are integers. Then
2p
2p − q
 2  2a − 1  q − 1  qs
is again a rational number.
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Conclusion
In addition to verifying how some properties that held for stochastic matrices in the
field of real numbers held in the finite field as well, we have proved analytically why in
fact they did or did not hold. Our main result being the property that 2  2 stochastic
matrices will have solutions to their characteristic equations that are in their respective
finite fields and as a consequence, the trace and determinant of any stochastic matrix
with a finite field equal the sum and product of its eigenvalues respectively. We have
also proven that we have only one trivial case for a linearly dependent row stochastic
matrix in the finite field as well. On the other hand, we have proved, mainly through
counterexample, that some properties that held in the real field did not hold in the finite
field. We have found that the spectral radius is not less than or equal to |1| and that
stochastic matrices whose entries are all strcictly positive do not necessarily have   1
as a simple eigenvalue. We have also proved why the properties we have derived as we
explored such matrices in the real field hold in an infinite subfield (the rational
numbers) of the real field and have laid the foundation for future work by extending our
scope to 3  3 stochastic matrices.
References
[1] Leon, Steven, Linear Algebra, 7th Edition, Prentice Hall, Upper Saddle River, New
Jersey, 2006
[2] Stewart, William, Introduction to the Numerical Solutions of Markov Chains, Princeton
University Press, Princeton, New Jersey,1994
[3] Hillier, Frederick, Introduction to Operations Research, McGraw Hill, New York, New
York, 2005
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