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Prerequisite links
Supplemental links
A Matrix and its Eigenvector
The reason we are stressing
eigenvectors and their importance is
because the action of a matrix
AA
on one of its eigenvectors
vv is
-
extremely easy (and fast) to calculate
Av=λv
A
v
λ
v
(1)
just multiply vv by
λ λ.
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easy to interpret: AA just
scales vv, keeping its direction
constant and only altering the vector's length.
If only every vector were an eigenvector of
AA....
Using Eigenvectors' Span
Of course, not every vector can be ... BUT ... For certain
matrices (including ones with distinct eigenvalues,
λλ's), their eigenvectors
span
ℂn
n
, meaning that for
any
x∈ℂn
x
n
, we can find
α
1
α
2
α
n
∈ℂ
α
1
α
2
α
n
such that:
x=
α
1
v1+
α
2
v2+…+
α
n
vn
x
α
1
v
1
α
2
v
2
…
α
n
v
n
(2)
Given
Equation 2, we can rewrite
Ax=b
A
x
b
. This equation is modeled in our LTI system
pictured below:
x=∑i
α
i
vi
x
i
α
i
v
i
b=∑i
α
i
λ
i
vi
b
i
α
i
λ
i
v
i
The LTI system above represents our
Equation 1. Below is an illustration of the steps taken
to go from
xx to
bb.
x→α=V-1x→ΛV-1x→VΛV-1x=b
x
α
V
-1
x
Λ
V
-1
x
V
Λ
V
-1
x
b
where the three steps (arrows) in the above illustration represent
the following three operations:
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Transform x x
using
V-1
V
-1
- yields
αα
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Action of AA in new basis -
a multiplication by Λ
Λ
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Translate back to old basis - inverse transform using a
multiplication by V V, which
gives us bb
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"My introduction to signal processing course at Rice University."