Eigenvalues of 2x2 Eigenvalues of 2x2 Matrices overMatrices over

By Benjamin CarrollBy Benjamin CarrollpZ

ObjectivesObjectives

►We will study eigenvalues of 2x2 We will study eigenvalues of 2x2 matricesmatrices

in . in . ►We will investigate conditions under We will investigate conditions under

which eigenvalues of A are in .which eigenvalues of A are in .►We will discuss the discriminant as the We will discuss the discriminant as the

form of an ellipse, show where form of an ellipse, show where eigenvalues exist, and some eigenvalues exist, and some corresponding matrices.corresponding matrices.

Zp

Zp

Zp

Question!!

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

Let A a b

c dwhere a,b,c and d are in Zp

where p is a prime number.

Eigenvalues of Eigenvalues of AA

The characteristic polynomial of A:The characteristic polynomial of A:

Eigenvalues of A:Eigenvalues of A:

wherewhere

Eigenvalues of A in exist for p>2 if and only if: Eigenvalues of A in exist for p>2 if and only if:

1 , 2 2 1 a d a d2 4ad bc

2 a d ad bc

Zp22 )(4)( nbcadda

2 1 p 1

2,p 3

We see that although we will not have eigenvalues in .

Also, for , 2 in the denominator in is 0, and therefore, undefined.

p 2in Z2 :

a d ad bc P a d2 4ad bc 1 , 2

0 0 2 0 0,0

1 2 1 0 0,0

1 0 2 1 0,1

1 2 1, 3 1 no sol’n in Z2

a d 1 and ad bc 1

Z22 1 p 1

2

Z2

Eigenvalues of Eigenvalues of AA

Let Let

be the discriminant.be the discriminant.

Question!!!Question!!!For what values of a, b, c and d is D a perfect For what values of a, b, c and d is D a perfect

square (1, 4, 9, 16, 25, …) ?square (1, 4, 9, 16, 25, …) ?

bcdabcaddaD 4)()(4)( 22

Eigenvalues in !! Z3in Z3 :

a d ad bc P a d2 4ad bc 1 , 2

0 0 2 0 0,0

1 2 1 4 2 no sol’n in Z3

2 2 2 8 1 2,2

1 0 2 1 0,1

1 2 1, 3 0 2,2

2 2 2 7 2 no sol’n in Z3

2 0 2 2 4 1 0,2

1 2 2 1 0 1,1

2 2 2 2 4 2 no sol’n in Z3

Three Cases whereThree Cases where2nD

►We will look at three cases of the We will look at three cases of the discriminant (we will use discriminant (we will use ): ): Case 1:Case 1: Case 2:Case 2: Case 3:Case 3:

D M2

D 4N2

D M2 4N2

D a d2 4bc

Case 1:

Examples of A:

2nD

a d 3 0 or a d 3 1 or a d 3 2 4bc 3 0a 0,d 0 b 0,c 0a 1,d 1 b 0,c 1a 2,d 2 b 0,c 2

b 1,c 0b 2,c 0

D M2 a d M and 4bc 0

0 0

0 0

1 0

1 0

1 2

0 2

Case 2:

Examples of A:

2nD

D 4N2 a d 0 and 4bc 4N2

a d 3 0 4bc 3 0 or 4bc 3 1a 0,d 0 b 0,c 0 b 1,c 1a 1,d 1 b 0,c 1 b 2,c 2a 2,d 2 b 0,c 2 b 1,c 2

b 1,c 0 b 2,c 1b 2,c 0

2nD

1 1

1 1

2 1

0 2

Case 3:

Examples of A:

2nD D M2 4N2 a d M and 4bc 4N2

0 0

0 0

1 1

1 1

1 0

1 0

1 2

0 2

a d 3 0 4bc 3 0a d 3 0 4bc 3 1a d 3 1 4bc 3 0a d 3 2 4bc 3 0

Eigenvalues in !!Zp

in Zp :

for any 2 2 matrix where a d2 4bc p 0,1,4,9,16, . . . eigenvalues will exist in Zp .

In other words, where:Case 1: D M2 a d p M 4bc p 0

Case 2: D 4N2 a d p 0 4bc p 4N2

Case 3: D M2 4N2 a d p M 4bc p 4N2

Discriminant/EllipseDiscriminant/Ellipse

►The discriminant from Case 3 The discriminant from Case 3

is a variant of the equation of an is a variant of the equation of an ellipse!!ellipse!!

D M2 4N2 n2

M2

n 2 N2

n2

4

1

D M2 4N2 n2

M, 2N,n forms a Pythagorean triple

a d M and bc N

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

Pythagorean Triples A,B,C for A2 B2 C2

B jk, A j2 k 2 , and C j2 k 2 where j k

Solutions M, 2N,nare of forms:

2N 2jk or N jk, M j2 k 2 , and n j2 k 2 They are listed as follows:j k 2N M n2 1 4 3 53 1 6 8 103 2 12 5 13....

Graphs of the EllipsesGraphs of the Ellipses

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

11

12

M2 4N2 n2

(M,2N,n) (M,2N,n)

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

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??!!

D M2 4N2 n2

D 32 422 25 52

M2 4N2 n2

Patterns for EllipsesPatterns for Ellipses

►M=3, N=2, n=5. This is true for all M=3, N=2, n=5. This is true for all Pythagorean Triples (M,2N,n) of the Pythagorean Triples (M,2N,n) of the natural numbers in the 1natural numbers in the 1stst quadrant quadrant where:where: M is a multiple of 3, then multiplied by 2 for M is a multiple of 3, then multiplied by 2 for

each ellipse equation,each ellipse equation, N is a multiple of 2, then multiplied by 2 for N is a multiple of 2, then multiplied by 2 for

each ellipse equation, andeach ellipse equation, and n is a multiple of 5, then multiplied by 2 for n is a multiple of 5, then multiplied by 2 for

each ellipse equation. each ellipse equation.

M2 4N2 n2

Patterns for Ellipses

32g2 422g2 52g2

M2 4N2 n2

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.

3M2 42N2 5n2 forM,N,n 2g

where g 0,1,2,3, . . . represents each graph (0 being the inner-most ellipse)

Therefore, we can further simplify the equation as:

Patterns for EllipsesPatterns for Ellipses

►Here are some examples for this Here are some examples for this equation:equation:

M2 4N2 n2

g 32g2 422g2 52g2 M N n M2 4N2 n2

03 2 422 5 2

3202 42202 5202 3 2 5 32 422 52

16 2 442 10 2

3212 42212 5212 6 4 10 62 442 102

212 2 482 20 2

3222 42222 5222 12 8 20 122 482 202

324 2 4162 40 2

3232 42232 5232 24 16 40 242 4162 402

... ... ... ... ... ...

Perfect Squares in EllipsesPerfect Squares in Ellipses

► Looking at the 2Looking at the 2ndnd, 3, 3rdrd, and 4, and 4thth quadrants, quadrants,

the P.T.’s are:the P.T.’s are:

(-M,2N,n)=(p-M,2N,n),(-M,2N,n)=(p-M,2N,n),

(-M,-2N,n)=(p-M,2(p-N),n),(-M,-2N,n)=(p-M,2(p-N),n),

(M,-2N,n)=(M,2(p-N),n), (M,-2N,n)=(M,2(p-N),n),

respectively, and form perfect squares in respectively, and form perfect squares in . But there is a stipulation!!! . But there is a stipulation!!!

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Perfect Squares in Ellipses

2nd Quadrant:

3rd Quadrant:

4th Quadrant:

p M2 2N2 n2

p M2 2p N2 n2

M2 2p N2 n2

7 62 442 1 64 65 7 2n2 102 100 7 2

7 62 47 42 1 36 37 7 2n2 102 100 7 2

62 47 42 36 36 72 7 2n2 102 100 7 2

in Z7 :

Matrices in All 4 Quadrants

1st Quadrant:

2nd Quadrant:

3,2 M2 4N2 n2 32 422 52

a d 5 3 4bc 422 5 1

4 1

2 1

3 3

4 0

0 2

1 2

3,2 2,2 p M2 4N2 n2 22 422 5 0n2 25 5 0

a d 5 2 4bc 422 5 1

4 1

2 2

3 2

1 1

1 4

3 4

Matrices in All 4 Quadrants3,2 2,3 p M2 4p N2 n2 22 432 5 0

n2 25 5 0a d 5 2 4bc 422 5 1

4 1

2 2

3 2

1 1

1 4

3 4

3,2 3,3 M2 4p N2 n2 32 432 5 0n2 25 5 0

a d 5 3 4bc 422 5 1

4 1

3 1

3 3

1 0

2 2

4 4

3rd Quadrant:

4th Quadrant:

Perfect Squares in EllipsesPerfect Squares in Ellipses

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

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Future WorkFuture Work

► Eigenvalues of 3x3 matrices in . Eigenvalues of 3x3 matrices in . ► Find condition(s) under which A has Find condition(s) under which A has

eigenvalues over extended fields .eigenvalues over extended fields .

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Questions????Questions????