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Eigenvalues of 2x2 Matrices over

Eigenvalues of 2x2 Matrices over. By Benjamin Carroll. Objectives. We will study eigenvalues of 2x2 matrices in . We will investigate conditions under which eigenvalues of A are in .

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Eigenvalues of 2x2 Matrices over

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  1. Eigenvalues of 2x2 Matrices over By Benjamin Carroll

  2. Objectives • We will study eigenvalues of 2x2 matrices in . • We will investigate conditions under which eigenvalues of A are in . • We will discuss the discriminant as the form of an ellipse, show where eigenvalues exist, and some corresponding matrices.

  3. Question!! For what values of a, b, c, and d do eigenvalues of A in exist?

  4. Eigenvalues of A The characteristic polynomial of A: Eigenvalues of A: where Eigenvalues of A in exist for p>2 if and only if:

  5. We see that although we will not have eigenvalues in . • Also, for , 2 in the denominator in is 0, and therefore, undefined.

  6. Eigenvalues of A Let be the discriminant. Question!!! For what values of a, b, c and d is D a perfect square (1, 4, 9, 16, 25, …) ?

  7. Eigenvalues in !!

  8. Three Cases where • We will look at three cases of the discriminant (we will use ): • Case 1: • Case 2: • Case 3:

  9. Case 1: Examples of A:

  10. Case 2: Examples of A:

  11. Case 3: Examples of A:

  12. Eigenvalues in !!

  13. Discriminant/Ellipse • The discriminant from Case 3 is a variant of the equation of an ellipse!!

  14. Ellipses and Pythagorean Triplesfrom Wikipedia http://en.wikipedia.org/wiki/Pythagorean_triples

  15. Graphs of the Ellipses

  16. (M,2N,n) • For all possible Pythagorean Triples of the form (M,2N,n), all values of M exist on the x-axis, all values of N exist on the y-axis, and all values of n exist on the x-axis on the ellipse of the intersection of M and N.

  17. Patterns for Ellipses There does exist LOTS of patterns for finding these P.T.’s!!!! For example: • Lets take a look at the point (3,2) and our new discriminant formula Notice a pattern yet??!!

  18. Patterns for Ellipses • M=3, N=2, n=5. This is true for all Pythagorean Triples (M,2N,n) of the natural numbers in the 1st quadrant where: • M is a multiple of 3, then multiplied by 2 for each ellipse equation, • N is a multiple of 2, then multiplied by 2 for each ellipse equation, and • n is a multiple of 5, then multiplied by 2 for each ellipse equation.

  19. Patterns for Ellipses • A more general look for any pattern would be to look at P.T.’s of the form (rM,2sN,tn), where M is a multiple of r, N is a multiple of s, and n is a multiple of t.

  20. Patterns for Ellipses • Here are some examples for this equation:

  21. Perfect Squares in Ellipses • Looking at the 2nd, 3rd, and 4th quadrants, the P.T.’s are: (-M,2N,n)=(p-M,2N,n), (-M,-2N,n)=(p-M,2(p-N),n), (M,-2N,n)=(M,2(p-N),n), respectively, and form perfect squares in . But there is a stipulation!!!

  22. Perfect Squares in Ellipses 2nd Quadrant: 3rd Quadrant: 4th Quadrant:

  23. Matrices in All 4 Quadrants 1st Quadrant: 2nd Quadrant:

  24. Matrices in All 4 Quadrants 3rd Quadrant: 4th Quadrant:

  25. Perfect Squares in Ellipses • Now that we have established that perfect squares exist in each quadrant for the equation of ellipses in , we know where there will be eigenvalue solutions.

  26. Future Work • Eigenvalues of 3x3 matrices in . • Find condition(s) under which A has eigenvalues over extended fields .

  27. Questions????

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