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The Manga Guide to Linear Algebra by Ltd. Trend-Pro Co., Iroha Inoue, Shin Takahashi

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What Are Eigenvalues and Eigenvectors? 211
What Are Eigenvalues and
Eigenvectors?
What do you say we
start o with a few
problems?
Sure.
H...
Like this?
So close!
Oh, like
this?
Exactly!
So...the answer can
be expreed using
multiples of the
original two vectors?
Okay, first problem.
Find the image of
using the linear
transformation
determined by the
2 matrix
3
1
1
2
c
1
+ c
2
8
2
3
1
(where c
1
and c
2
are real numbers).
Like so.
Oh...
212 Chapter 8 Eigenvalues and Eigenvectors
That’s right! So you
could say that the
linear transformation
equal to the matrix
8
2
3
1
...transforms All points
on the
x
1
x
2
plane...
Like this?
Coect.
So this solution can
be expreed with
multiples as we...
H
Let's move on to another problem.
Find the image of using
(where
c
1
,
c
2
, and
c
3
are real numbers).
c
1
+ c
2
+ c
3
1
0
0
0
1
0
0
0
1
4
0
0
0
2
0
0
0
−1
the linear transformation
determined by the
3×3
matrix
What Are Eigenvalues and Eigenvectors? 213
214 Chapter 8 Eigenvalues and Eigenvectors
4
0
0
0
2
0
0
0
−1
...transforms every
point in the
x
1
x
2
x
3
space...
So you could
say that the
linear transformation
equal to the matrix
Like this.
I get
it!
4
0
0
0
2
0
0
0
−1
...transforms every
point in the
x
1
x
2
x
3
space...
So you could
say that the
linear transformation
equal to the matrix
Times 4
Lking
back
Times 2
kping those
examples in mind.
Eigenvalues and eigenvectors
a
11
a
21
a
n1
a
12
a
22
a
n2
a
1n
a
2n
a
nn
x
1
x
2
x
n
x
1
x
2
x
n
x
1
x
2
x
n
If the image of a vector through the linear transformation determined by the matrix
is equal to
λ
,
λ
is said to be an eigenvalue to the matrix,
and is said to be an eigenvector corresponding to the eigenvalue
λ
.
The zero vector can never be an eigenvector.
R
n
R
n
x
1
x
2
x
n
λ
x
1
x
2
x
n
So the two
examples could
be suarized
like this?
Exactly!
You can generay
never find more than n
dierent eigenvalues
and eigenvectors for
any n×n matrix.
Oh...
g
s
an
d
Ei
ge
nv
e
c
t
o
rs
Let's have a
lk at the
definition...
Matrix
Eigenvalue
Eigenvector
4
0
0
0
2
0
0
0
−1
8
2
3
1
λ = 7, 2 λ = 4, 2, −1
the vector
corresponding
to
λ = 7
3
1
the vector
corresponding
to
λ = 2
1
2
the vector
corresponding
to
λ = 4
1
0
0
the vector
corresponding
to
λ = 2
0
1
0
the vector
corresponding
to
λ = 1
0
0
1

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