Tuesday, November 03, 2009
Eigen Vectors
**
=
+
x
Ax bt
=
x0
x0
**
Eigenvalues/eigenvectors
=
Awi
λiwi

Eigenvalues/Eige vectors of same or different values form linear independent
sectors

Each Eigenvalue λ
i
has at least one Eigenvector

If m
i
is the multiplicity of λ
i
it is positive then λ
i
has 1, 2, 3… or m
i
eigenvectors
Ex3.
=
 →
A
230 2
only one eigenvector
Ex4.
=

A
200 2
→
2 eigenvector
*If there are as many Eigenvectors as the size of the matrix, then there is a
full set
of the
eigenvector
nxn matrix A has n eigenvectors
Assume a full set of eigenvectors:
…
=
…
AW1W2
Wn
W1W2
Wn λ1000λ2000λn


(eigenvector matrix)(eigenvalue matrix)
In a 2x2 case
=
→
=
AW1W2
W1W2λ100λ2
a11a12a21a22w11w12w21w22
w11w12w21w22λ100λ2
→
+
+
+
+
=
a11w11 a12w21a11w12 a12w22a21w11 a22w21a21w12 a22w22
λ1w11λ2w12λ1w21λ2w22
=
→
…
=
…
→
=
Aw1
λiwi
AW1W2
Wn
W1W2
Wnλ1000λ2000λn
AW
WΛ
Multiply by W
1
if W is nxn matrix (full set of eigen vector)
AWW
1=
W
1
WΛ
→
A
=
Λ
Back to ** in to top box
Define
xt
=
Wzt
→
zt
=
W
1
x
(
t
)
1
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Tuesday, November 03, 2009
=
+
→
=
+
;
}
=
ddtWz
AWz bt
Wz
AWz bt initial conditions change
Wz0
x0
=
=
ddta ft
adfdt
ddtA ft
Adfdt

=

+

→ =
+ ( )
W 1Wz
W 1AWz W 1bt
z
Λz ξ t



I
Λ
ξ
=
+
(
) 
zit
zi0eλit eλit0tξi t' e λit'
If b
(t)=constant
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 Fall '08
 Hawkins
 Eigenvalue, eigenvector and eigenspace, eigenvector nxn matrix

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