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Find EIGENVALUES and EIGENVECTORS for the matrix: 4 3 6 5 A 4 3 6 5 0 0 A I 4 3 6 5
| Date post: | 01-Jan-2016 |
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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.
