Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Finishing Eigenvalues
You can also have COMPLEX roots
...

[
]
2 1
Example: Find the eigenvalues/vectors of

...
Factoring:
√
2± 4−8
2 ± 2i
λ=
=
= 1 ± i
...

[
[
]
]
1
1−i
1
0
1
(1 + i) 0
2
λ=1+i:
−→
−2 −1 − i 0
1−i
1
0
[
]
[
]
1 1 (1 + i) 0
1+i
2
eventually breaking down into
for a vector

...

for a vector 2
or
2
0
1

Property: Complex eigenvalues from REAL matrices come in conjugate pairs, obviously,
but so do their associated eigenvectors
...
You need to use zw = zw and understand that it works for
matrix multiplication
...
That
equation makes v an eigenvector for value λ
...

So, really, what we’re looking for when finding eigenvectors is bases to the eigenspaces
...
The
whole set will be L
...
as well if the vectors from individual eigenspaces are kept linearly
independent
...
If that set spans Rn (with A n × n) then you have a potentially
helpful property
...

2 −1

det(A − λI) =

1−λ
4
2 −1 − λ

= (1 − λ)(−1 − λ) − 8
= λ2 − 1 − 9
= λ2 − 9 = (λ − 3)(λ + 3)

so the eigenvalues are ±3
...

1
[
[
]
]
4 4 0
1 1 0
λ2 = −3 =⇒
−→
2 2 0
0 0 0
]
[
−1
or any non-zero multiple
...

1
1
]
[
1
1 1
−1
P =

...

3 −3
0 −3
0 λ2
3 −1 2
3 0 −9

It’s a diagonal matrix composed of the (ordered) eigenvalues
...

Definition: The matrix A, n × n, is diagonalizable if you can write
P DP −1 = A

P −1 AP = D

with D a diagonal matrix
...


2

Note: being diagonalizable is a totally disconnected property from being invertible
...
However, its characteristic polynomial is λ2 − 2λ + 1,
0 1
0
1
for a double λ = 1 eigenvalue
...

0
0 0 0
It is missing a second eigenvector, so it’s invertible but not diagonalizable
...
It is, however, diagonaliz0 0
able, with eigenvalues 1 and 0
...

•
0 1
[
]
0 1
•
is not invertible, and not diagonalizable
...

Property: A is not diagonalizable if: its characteristic equation has a factor (λ − b)k for
eigenvalue b while (A − bI) has fewer than k free variables
...

Note that A can be diagonalizable if it has double, triple, etc, root in its characteristic
equation
...

A question of this sort involves
• Determining if the matrix is diagonalizable (does it have enough eigenvectors)
...

• Writing them out in the correct order, so D = P −1 AP or A = P DP −1
...
Also, make
sure you’ve got P , D and P −1 in the same order (there is no right order, but they have to
be consistent)
...
Notice:
(
)2
A2 = P DP −1 = P DP −1 P DP −1 = P D2 P −1
...
How does

λk 0 0
1
 0 λk 0
2

 0 0 λk
k
D =
3

...


...


...


...

· · · λk
n

so these are pretty much the easiest matrices to calculate powers
...

Also: those who are continuing into science will probably have to deal with systems of
differential equations:
∂
y = Ay
=⇒
y = eAt y(0)
...

=P



...


...


...


...
We’ll first take a look at a more
helpful variant
...
, yn } is an orthonormal set if the set is orthogonal
and |y1 |2 = y1 · y1 = 1
...
, xn } and use |x1 | , |x2 | ,
...

Definition: A matrix is orthogonal if it has columns (or rows) composed of an orthonormal
set
...
For example, look at it

0 0
1 0 
0 1

using the orthonormality
...

Properties:
• P orthogonal then det(P ) = ±1
...

• P , Q, both orthonormal then P Q is as well
...

Definition: A is orthogonally diagonalizable if P T AP = D, with D a diagonal matrix and
P T P = I
...

−2
1
Notice that AT = A
...


5

Theorem:(Principle Axis)
The following are equivalent for A, an n × n matrix:
• A is orthogonally diagonalizable
• A has a set of eigenvectors that form an orthonormal basis of Rn
• AT = A (so A is symmetric)
...
If A is orthogonally
diagonalizable then:
A = P DP T

=⇒

(
)T
AT = P DP T = P DT (P T )T = P DP T = A,

so A has AT = A and is symmetric
...
There’s one extra element
...
When you find an eigenvalue with multiple roots in the characteristic equation,
you’ll have to use Gram-Schmidt on the resulting vectors
...

Example: Diagonalize  −1
1 −1
2
Exercises:
Section 2
...
bdf), 6
...
bdfh), 8, 17
...
7: 2
...
bc) (c is messy)

6



---