1. By Josh Zimmer Department of Mathematics and Computer Science The set ℤ p = {0,1,...,p-1} forms a finite field. There are p ⁴ possible 2×2 matrices in ℤ p. 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, ℤ p 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 ℤ p A row (column) stochastic matrix is a matrix whose row sums (column sums) are equal to a constant k in ℤ p. A doubly stochastic matrix is both row and column stochastic. Let A r, A c and A d be respectively, row, column, and doubly stochastic matrices in ℤ p. Then: Eigenvalues are of the form: A 2×2 rank one matrix is of the form: A nilpotent matrix has the property A k =0 for k>0. It is known in the real field, nilpotent matrices are of the form: We find nilpotent matrices in ℤ p are: Real symmetric matrices (A T =A) always have eigenvalues in the real field. Let Skew-symmetric matrix is where A T =-A. In this case, the off- diagonal elements of A are not the same, but are the additive inverse (in ℤ p ) of each other. A is orthogonal if AA T =A T A=k²*I for k² in ℤ p. In the real field, eigenvalues of an orthogonal matrix are 1 and -1 All eigenvalues of rank-one matrices are of the form: λ 1 = v T u, λ 2 = 0 and are therefore in ℤ p λ 2 = trace of A – λ 1 All eigenvalues of nilpotent matrices are zero mod(p). For p = 2 Eigenvalues exist in ℤ p iff: Examples: (1) b=1 In this project, we study special structured 2x2 matrices in ℤ p. There are at most, 2p 3 +p 2 stochastic matrices, p 4 rank-one matrices, 3p nilpotent matrices, p 3 symmetric matrices, p skew-symmetric matrices, and p 2 symmetric orthogonal matrices in ℤ p. Due to the construction of stochastic, rank-one, and nilpotent matrices in ℤ p, they will always have eigenvalues in ℤ p. We have derived conditions, respectively, for symmetric, skew- symmetric, and symmetric orthogonal matrices, under which eigenvalues are in ℤ p. Currently, we are studying the eigenvalue properties of non- symmetric orthogonal matrices and other special structured matrices. Let λ 1 = k, (1) (2) Note: non-symmetric orthogonal matrices are still being investigated (1) (2) Examples: For p>2 where u and v are non-zero column vectors in ℤ p. There are p 4 possible rank one matrices in ℤ p. There are only p possible skew-symmetric matrices in ℤ p (2) There are at most 3p possible nilpotent matrices in ℤ p. There are at most p 2 orthogonal matrices in ℤ p There are at most 2p 3 +p 2 stochastic matrices in ℤ p. There are p 3 possible symmetric matrices in ℤ p Skew-symmetric matrices in ℤ p : b Ý p ? b Þ ¯ p m 2 123 Symmetric orthogonal matrices are: Eigenvalues are λ 1 =k and λ 2 =-k. λ1 and λ2 are in ℤ p iff (a, b, k) are Pythagorean Triples P(λ)=λ 2 -(a+d) λ+(ad-b 2 ) λ 1, λ 2 in ℤ p iff D(a.b.b.d) = (a-d) 2 +4b 2 =n 2 <p iff (a-d, 2b, n) is a Pythagorean Triple