Special Square Matrices (2x2)
over Zp
By OC Josh Zimmer
References Used
Linear Algebra with Applications 7E.
Leon, Steven.
Discrete Mathematical Structures 5E.
Kolman, Busby, Ross.
Linear Algebra with Applications 5E.
Strang, Gilbert.
Today
Which Finite Fields
List Special Matrices
List Properties
Give Examples
What we are looking for
Different ways how to find it
Matrices in Zp





Z2 = {0, 1}
Z3 = {0, 1, 2}
Z5 = {0, 1, 2, 3, 4}
Z7 = {0, 1, 2, 3, 4, 5, 6}
Zp where p is a prime number
Types of Special Square Matrices
Symmetric, Skew-symmetric matrices
Orthogonal matrices
Nilpotent, Idempotent matrices
Stochastic matrices
Rank One matrices
What makes a Special Square Matrix
 Obviously square (2x2)
 Types of special we are concerned with





Symmetric, Skew-symmetric matrices
Orthogonal matrices
Nilpotent, Idempotent matrices
Stochastic
Rank One
 First starting with small finite fields (Z2)
then moving higher
Symmetric Matrices
A
a b
c d
A
a b
b d
where a 21 a 12 b c
therefore A T A
Some Examples
 Z3
2 1
1 0
1 0
0 2
 Z5
3 4
 Z7
0 5
4 1
5 2
Skew-Symmetric Matrices
A
a b
and A T -A
c d
a c
b d

a b
c d
Therefore a -a
 a 0
d -d
 d 0
and c -b
 c p b
Some Examples
3 : A 
5 : A 
0 1
2 0
0 1
4 0
Orthogonal Matrices
A
a b
c d
A A
T
a c
b d
ab cd p 0
ac bd p 0
a 2 c 2 p k
b 2 d 2 p k
a 2 b 2 p k
c 2 d 2 p k
where A T A AA T k I N

a b
c d

a 2 c 2 ab cd
ac bd b 2 d 2

a 2 b 2 ac bd
ac bd c 2 d 2
AA T
Some Examples
3 :
1 0
0 1
0 1
1 0
1 0
0 1
Nilpotent Matrices
A
k 2
a b
A k 0
c d
A2 
a 2 bc p
ab bd p
ac cd p
bc d 2 p
0
0
0
0
a b
c d

a b
c d

a 2 bc ab bd
ac cd bc d 2
Nilpotent Matrix Examples
2 :
3 :
5 :
0 1
0 0
0 0
2 0
0 0
3 0
Idempotent Matrices
A
a b
c d
k 2
A k Afork 2
A2 
a b
c d

a 2 bc p a
ab bd b
a dp b
ac cd c
a dp c
bc d 2 p d
a b
c d

a 2 bc ab bd
ac cd bc d 2
Idempotent Matrix Examples
3 :
1 0
0 1
3 :
1 0
1 0
Stochastic Matrices
Properties
Each row and/or column sum = 1 or =k
λ1 = k, |λi| < k, k in Zp
Examples
1 
2 0
0 2
2 1
1 2
2 1
1 2
2 0
0 2
1
1
in 
1 k 1
for what a b p k
and b 
d p k
Rank One Matrices
Properties
A = u vt
λ = vt u or 0
Examples
A u v
A
T
a 2 ac
ac c 2
u
a 0
c 0
,
v 
T
a c
0 0
Real Eigenvalues
 Under what conditions do real
eigenvalues exist
 A2*2 over Z2 has 16 different possible
matrices
 Eigenvalues of these matrices
 of A2*2 in Z3 over Z3
 λ²-(a+d)λ+(ad-bc)=0
Ones that Exist
An example where eigenvalues exist in Z3 :
 2 a d ad b 2  where a d 1, ad b 2 0
 2 
to find the eigenvalues we use the quadratic formula
b b 2 4ab
2a
2
 1 , 2
where, in the characteristic polynomial, a d is b and ad b is b and 1 is a.

a
d

a
d2 4
1

adb 2 
2
1

1 
12 4
1

0
2
1

1 1
2
2
 1 0, 2 2 3 1
clearly, eigenvalues exist in Z3 , for 

  .
What to look for
a b c 0 b2 b 2 4ac
2a
2
Discriminant 
a d2 4
ad b 2 

a d 4b
2
So when is 2 
a d2 4b 2 in p
when 
a d2 4b 2 is a perfect square
2
How do we find these Eigenvalues?
in Z3 :
"B"
"C"
Polymonial
a d ad b 2 P

Discriminant
Eigenvalues

a d2 4
ad b 2  1 , 2
0
0
2
0
0, 0
0
1
2 1
4 2
no sol’n in Z3
0
2
2 2
2, 2
1
0
2 
0, 1
1
1
2  1,
2, 2
1
2
2  2
2
0
2 2
0, 2
2
1
2 2 1
1, 1
2
2
2 2 2 4 2
no sol’n in Z3
7 2
no sol’n in Z3
Note: for 2  1, there is no solution for 
a d2 4
ad b 2 except under Z3 .
Properties
 Eigenvalues
Special Matrix
Symmetric
Skew-Symmetric
Stochastic = k
Orthogonal
Nilpotent
Idempotent
Rank One
Properties in R Properties in Zp
λi in R
λi in Zp
λi = 0 or C
λi = 0 or C
λ1 = k, |λi|<k
λi = |1|
λi = 0
λi = 0 or 1
λi = vtu
λ1 = k, k in Zp
λi = 1 or -1= p-1
λi = 0
λi = 0 or 1
λi in Zp
What happens if they don’t Exist?
An example where eigenvalues do not exist in Z3 :
2 a d ad b 2  where a d 2, ad b 2 2
b b 2 4ab
2a
 2 2
2

a
d

a
d2 4
1

adb 2 
2
1

2 
22 4
1

2
2
1
 1 , 2

2 48
2

2 4
2

2 2
2
no sol’n in Z3
4 is 2 in Z3 and is clearly not a perfect square. Therefore, 

2 2 2 has
no eigenvalues in Z3 .
Limits due to field Zp
Matrix combinations where there were no eigenvalue solutions in p
or where there was only a unique solution in that field
2x2 matrices in Zp
3
No Solns 3
5
No Solns 10
7
No Solns 26
unique Solns 1
unique Solns 4
unique Solns 12
Remember
2  1 has no solution for 
a d2 4b 2 except under 3
How else to find them
How do we know when the discriminant is
a perfect square?
Pythagorean triples help us identify what
combinations will yield a perfect square
thus giving us an eigenvalue in Zp
Pythagorean Triples
A review of Pythagorean Triples 
A, B, Cfor A 2 B2 C 2 are of the forms:
B jk, A j 2 k 2 , and C j 2 k 2 where j k
We will apply this result to the equation M2 4N 2 n 2 . Solutions 
M, 2N, n
are of forms:
2N 2jk or N jk, M j 2 k 2 , and n j 2 k 2 They are listed as follows:
j
k 2N M n
2 1 4
3
5
3 1 6
8
10
3 2 12 5
13
4 1 8
15 17
4 2 16 12 20
4 3 24 7
25
Examples
EXAMPLES:
(3, 4, 5) => (j, k) = (1, 2)
(5, 12, 13) => (j, k) = (2, 3)
(8, 15, 17) => (j, k) = (1, 4)
Solve
We will look at what type of 2x2 nonsymmetric matrices give a Pythagorean Triple in this form. One
way to find this is to look at the discriminant equation itself. It is of the form of an ellipse, so lets take a
look at the graphs of several ellipses to get a clue for the Triples.
2
M
2
n
2
n2 1
N
4
Graph
14
13.5
13
12.5
12
11.5
11
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.50
-14
-13.5
-13
-12.5
-12
-11.5
-11
-10.5
-10
-9.5
-9-8.5
-8-7.5
-7-6.5
-6-5.5
-5-4.5
-4-3.5
-3-2.5
-2-0.5
-1.5
-1-0.500.5
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
6 6.5
7 7.5
8 8.5
9 9.5
1010.5
1111.5
1212.5
1313.5
14
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
-9.5
-10
-10.5
-11
-11.5
-12
-12.5
-13
-13.5
-14
cos
t
, 12 sin
t
Just looking at the first quadrant
N
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
M
Next…
 Making the connections
 Moving on to Z5, Z7, Zp
 When is the Discriminant a perfect
square?
Connections
 What are the relationships between each
matrix in Zp?
 What are the relationships between their
Eigenvalues?
 Are the Eigenvalues still in Zp?
Questions…
 Time ≤ 25 minutes