Special Square Matrices over a Finite Field
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Transcript Special Square Matrices over a Finite Field
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 dp b
ac cd c
a dp 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
d2 4
1
adb 2
2
1
1
12 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 b2 b 2 4ac
2a
2
Discriminant
a d2 4
ad b 2
a d 4b
2
So when is 2
a d2 4b 2 in p
when
a d2 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 d2 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 d2 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
d2 4
1
adb 2
2
1
2
22 4
1
2
2
1
1 , 2
2 48
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 d2 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, Cfor 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