Math 2331
Linear Algebra
Section 6.1
Eigenvalues and Eigenvectors
Consider a function
. What do the elements in the domain look like?
(
)
:
n
f
x
→
\
\
n
⎞
⎟
What do the elements in the range look like?
So, we can think of any vector x as the input to a function and Ax as the output. What
does Ax do to a vector?
Example: Let
1
2
0
1
A
⎛
=
⎜
⎝
⎠
and find Ax for the following vectors
0
1
x
⎛
⎞
=
⎜
⎟
⎝
⎠
1
1
x
⎛ ⎞
=
⎜ ⎟
⎝ ⎠
1
3
x
⎛
⎞
=
⎜
⎟
⎝
⎠
1
3
x
−
⎛
⎞
=
⎜
⎟
⎝
⎠
1
0
x
⎛
⎞
=
⎜
⎟
⎝
⎠

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What is special about the last one?
Another Example -
1
4
2
3
A
⎛
⎞
=
⎜
⎟
⎝
⎠
Find Ax when x is-
0
1
x
⎛
⎞
=
⎜
⎟
⎝
⎠
1
0
x
⎛
⎞
=
⎜
⎟
⎝
⎠
1
1
x
⎛ ⎞
=
⎜ ⎟
⎝ ⎠
2
1
x
−
⎛
⎞
=
⎜
⎟
⎝
⎠
The fact that the function f(x) = Ax does not change the direction of certain vectors is
very significant.