Find EIGENVALUES and EIGENVECTORS for the matrix:

43

65A

43

65

0

0

AI

43

65

43

65

43

65det

0122

)18()4)(5(2

The EIGENVALUES are 2 -1and

63

63

43

65

20

02

2 AI

63

63

43

65

20

02

2 AI The NULL SPACE of

0

0

63

63

2

y

x

AI

contains solutions to:

The EIENVECTORS belonging to 2 are nonzero multiples of

1

2

33

66

43

65

10

01

1 AI

33

66

43

65

10

01

1 AI The NULL SPACE of

0

0

33

66

1

y

x

AI

contains solutions to:

The EIENVECTORS belonging to -1 are nonzero multiples of

1

1

has EIGENVALUES : 2 and -1and EIGENVECTORS:

43

65A

1

1

1

2and

Cosets of eigenspace??

has EIGENVALUES : 2 and -1and EIGENVECTORS:

43

65A

1

1

1

2and

Consider the EIGENSPACE W =

1

1

43

65A Consider the EIGENSPACE W =

1

1

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W

Add the same vector

to every point on W

to get a COSET OF W

43

65A Consider the EIGENSPACE W =

1

1

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W

Add the same vector

to every point on W

to get a COSET OF W - a line parallel to W

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43

65A Consider the EIGENSPACE W =

1

1

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The COSETS of the EIGENSPACE W are lines parallel to W.

43

65A Consider the EIGENSPACE W =

1

1

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W

The blue coset canbe obtained by adding the vector

a

0

to each vector in W

43

65A Consider the EIGENSPACE W =

1

1

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W

The blue coset canbe obtained by adding the vector

a

0

to each vector in W

43

65A Consider the EIGENSPACE W =

1

1

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W

The blue coset canbe obtained by adding the vector

a

0

to each vector in W

If v is a vector in the blue coset then

v = k +

1

1

a

0

43

65A Consider the EIGENSPACE W =

1

1

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WIf v is a vector in the blue coset then

v = k +

1

1

a

0

v

43

65A Consider the EIGENSPACE W =

1

1

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WIf v is a vector in the blue coset then

v = k +

1

1

a

0

Av = kA + A

1

1

a

0

A vectoron W

43

65A Consider the EIGENSPACE W =

1

1

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WIf v is a vector in the blue coset then

v = k +

1

1

a

0

Av = kA + A

1

1

a

0

A vectoron W

Every point on theblue coset has an image on the greencoset.

43

65A

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...............WIf v is a vector in the blue coset then

v = k +

1

1

a

0

Av = kA + A

1

1

a

0

A vectoron W

Consider the EIGENSPACE W =

1

1

If B is a coset of an eigenspace W then A maps every point on B onto G , another coset of W.