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Special Square Matrices (2x2) over Zp

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

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  1. Special Square Matrices (2x2) over Zp By OC Josh Zimmer

  2. References Used • Linear Algebra with Applications 7E. • Leon, Steven. • Discrete Mathematical Structures 5E. • Kolman, Busby, Ross. • Linear Algebra with Applications 5E. • Strang, Gilbert.

  3. Today Which Finite Fields List Special Matrices List Properties Give Examples What we are looking for Different ways how to find it

  4. 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

  5. Types of Special Square Matrices Symmetric, Skew-symmetric matrices Orthogonal matrices Nilpotent, Idempotent matrices Stochastic matrices Rank One matrices

  6. 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

  7. Symmetric Matrices

  8. Some Examples Z3 Z5 Z7

  9. Skew-Symmetric Matrices

  10. Some Examples

  11. Orthogonal Matrices

  12. Some Examples

  13. Nilpotent Matrices

  14. Nilpotent Matrix Examples

  15. Idempotent Matrices

  16. Idempotent Matrix Examples

  17. Stochastic Matrices • Properties • Each row and/or column sum = 1 or =k • λ1 = k, |λi| < k, k in Zp • Examples

  18. Rank One Matrices • Properties • A = u vt • λ = vt u or 0 • Examples

  19. 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

  20. Ones that Exist

  21. What to look for

  22. How do we find these Eigenvalues?

  23. Properties Eigenvalues

  24. What happens if they don’t Exist?

  25. Limits due to field Zp

  26. 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

  27. Pythagorean Triples

  28. Examples EXAMPLES: (3, 4, 5) => (j, k) = (1, 2) (5, 12, 13) => (j, k) = (2, 3) (8, 15, 17) => (j, k) = (1, 4)

  29. Solve

  30. Graph

  31. Just looking at the first quadrant

  32. Next… Making the connections Moving on to Z5, Z7, Zp When is the Discriminant a perfect square?

  33. Connections What are the relationships between each matrix in Zp? What are the relationships between their Eigenvalues? Are the Eigenvalues still in Zp?

  34. Questions… Time ≤ 25 minutes

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