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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
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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
Some Examples Z3 Z5 Z7
Stochastic Matrices • Properties • Each row and/or column sum = 1 or =k • λ1 = k, |λi| < k, k in Zp • Examples
Rank One Matrices • Properties • A = u vt • λ = vt u or 0 • Examples
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
Properties Eigenvalues
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
Examples EXAMPLES: (3, 4, 5) => (j, k) = (1, 2) (5, 12, 13) => (j, k) = (2, 3) (8, 15, 17) => (j, k) = (1, 4)
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