by josh zimmer department of mathematics and computer science the set ℤ p = {0,1,...,p-1} forms a...
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By Josh ZimmerDepartment of Mathematics and Computer Science
The set ℤp = {0,1,...,p-1} forms a finite field. There are p⁴ possible 2×2 matrices in ℤp. We will study matrices in special structures, such as: stochastic, rank-one, nilpotent, symmetric, skew-symmetric, and orthogonal.
Special structured real matrices such as stochastic, rank-one, symmetric, skew-symmetric, orthogonal, and nilpotent matrices, have many interesting properties when they are in the real field. When a special structured matrix is over a finite field, ℤp where p is a prime number, does it still have all properties as it does in the real field? In this paper, we study eigenvalue properties of 2×2 special structured matrices with entries in ℤp
A row (column) stochastic matrix is a matrix whose row sums (column sums) are equal to a constant k in ℤp. A doubly stochastic matrix is both row and column stochastic. Let Ar, Ac and Ad be respectively, row, column, and doubly stochastic matrices in ℤp. Then:
Eigenvalues are of the form:
Ar a k a
k d d, Ac
a k d
k a d, Ad
a k a
k a a
A 2×2 rank one matrix is of the form: A uv T u1v 1 u1v 2u2v 1 u2v 2
A nilpotent matrix has the property Ak=0 for k>0.
It is known in the real field, nilpotent matrices are of the form:
A 0 n
0 0or A
0 0
n 0where n
We find nilpotent matrices in ℤp are:
A a a
p a p aA
0 b
0 0and A
0 0
c 0and
Real symmetric matrices (AT=A) always have eigenvalues in the real field. Let
Skew-symmetric matrix is where AT=-A. In this case, the off-diagonal elements of A are not the same, but are the additive inverse (in ℤp) of each other.
A 0 b
p b 0, where b p A is orthogonal if AAT=ATA=k²*I for k² in ℤp. In the real
field, eigenvalues of an orthogonal matrix are 1 and -1
A a 0
0 p a
A a b
c dwhere a, b, c and d are in Zp
All eigenvalues of rank-one matrices are of the form: λ1 = vTu, λ2= 0 and are therefore in ℤp
λ2= trace of A – λ1
All eigenvalues of nilpotent matrices are zero mod(p).
For p = 2
bp b p m2 p
bp m2 b2
Eigenvalues exist in ℤp iff:
Examples: (1) b=1 Eigenvalues are 1 a, and 2 p a p a
In this project, we study special structured 2x2 matrices in ℤp. There are at most, 2p3+p2 stochastic matrices, p4 rank-one matrices, 3p nilpotent matrices, p3 symmetric matrices, p skew-symmetric matrices, and p2 symmetric orthogonal matrices in ℤp. Due to the construction of stochastic, rank-one, and nilpotent matrices in ℤp, they will always have eigenvalues in ℤp. We have derived conditions, respectively, for symmetric, skew-symmetric, and symmetric orthogonal matrices, under which eigenvalues are in ℤp. Currently, we are studying the eigenvalue properties of non-symmetric orthogonal matrices and other special structured matrices.
Let
λ1= k,
(1)
(2)
Note: non-symmetric orthogonal matrices are still being investigated
(1)
(2)
Examples:
1 , 2 2 1 a b a b2 4ad b2
For p>2
where u and v are non-zero column vectors in ℤp.There are p4 possible rank one matrices in ℤp .
p b bp b m 2 exist in p
3 0 0 0 Yes3 1 2 No3 2 2 No5 0 0 0 Yes5 1 4 2 Yes5 2 4 2 Yes5 3 4 2 Yes5 4 4 2 Yes
b m 2 m 2 1 primep
1 4 5 Yes1 16 17 Yes1 36 37 Yes1 64 65 No
There are only p possible skew-symmetric matrices in ℤp
(2)
There are at most 3p possible nilpotent matrices in ℤp .
There are at most p2 orthogonal matrices in ℤp
A a b
b p awhere b 0 a2 b2 p k 2
There are at most 2p3+p2 stochastic matrices in ℤp.
a d ad b2 P Soln of P 0
0 0 2 1 0, 2 0
0 1 2 1 1 1, 2 1
1 0 2 1 0, 2 1
1 1 2 1 No soln in 2
A 6 4
4 0A
10 6
6 1
1 1 2 5 1 2 2 9
A a b
b dThere are p3 possible symmetric matrices in ℤp
a d 2b b n3 4 2 58 6 3 105 12 6 1315 8 4 1712 16 8 20
7 11Skew-symmetric matrices in ℤp :
b p b p m2
1 A 3 p p 0, 2 A 3 a p a p 0
1 2 3
Symmetric orthogonal matrices are:
Eigenvalues are λ1=k and λ2=-k.
λ1 and λ2 are in ℤp iff (a, b, k) are Pythagorean Triples
P(λ)=λ2-(a+d) λ+(ad-b2) λ1, λ2 in ℤp iff D(a.b.b.d) = (a-d)2+4b2=n2<p iff (a-d, 2b, n) is a Pythagorean Triple