Properties of Special Square Matrices over a Field

1. 2 Ý Þ ? ¯ b p b m p Properties of Special Square Matrices over a Field By Josh ZimmerDepartment of Mathematics and Computer Science Introduction Abstract Stochastic 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: 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 Let There are at most 2p3+p2 stochastic matrices in ℤp. Nilpotent λ1= k, Eigenvalues are of the form: λ2= trace of A – λ1 Rank-one Symmetric Real symmetric matrices (AT=A) always have eigenvalues in the real field. Let 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 2×2 rank one matrix is of the form: There are p3 possible symmetric matrices in ℤp where u and v are non-zero column vectors in ℤp. For p = 2 There are p4 possible rank one matrices in ℤp . (1) All eigenvalues of rank-one matrices are of the form: λ1 = vTu, λ2= 0 and are therefore in ℤp P(λ)=λ2-(a+d) λ+(ad-b2) We find nilpotent matrices in ℤp are: Skew-Symmetric 1 2 3 (2) For p>2 There are at most 3p possible nilpotent matrices in ℤp . 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 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. All eigenvalues of nilpotent matrices are zero mod(p). Orthogonal Examples: Skew-symmetric matrices in ℤp : 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 There are only p possible skew-symmetric matrices in ℤp Symmetric orthogonal matrices are: (1) Eigenvalues exist in ℤp iff: Conclusions Examples: (1) b=1 (2) b>2 (2) 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. There are at most p2 orthogonal matrices in ℤp Eigenvalues are λ1=k and λ2=-k. λ1 and λ2 are in ℤp iff (a, b, k) are Pythagorean Triples Note: non-symmetric orthogonal matrices are still being investigated