Special Square Matrices (2x2) over Zp

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