by josh zimmer department of mathematics and computer science the set ℤ p = {0,1,...,p-1} forms a...

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