Linear Recurrence using matrices
Jonathan L.F. King
University of Florida, Gainesville FL 32611
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
26 October, 2011
(at
10:39
)
See also
Problems/NumberTheory/fibonacci.latex
Notation.
Use
“
r
∼
”
for
rowequivalence:
I.e,
K
×
N
matrices
M
r
∼
M
0
IFF
we can get from
M
to
M
0
using row operations.
Use “
c
∼
” for column
equivalence.
Two
N
×
N
matrices
B
&
C
are
similar
, or
con
jugate
to each other, if
there exists
an invertible
matrix
U
such that
U
1
·
B
·
U
=
C
.
Use
B
sim
C
for the similarity equivrelation.
Call a matrix OTForm
s
I
note
==== [
s
0
0
s
], a
dilation
;
its action on the plane is simply to scaleuniformly
by factor
s
.
Every
matrix commutes with
I
, and so:
A dilation is only conjugate to itself.
1
:
I.e, for invertible
U
, necessarily
U
1
·
s
I
·
U
=
s
I
.
For
T
a square matrix (
or a trn from a vectorspace to
itself
) and a complex number
α
, define the subspace
E
T
,α
:=
vectors
v
T
v
=
α
v
.
2
:
Saying that
“
α
is a
T
eigenvalue”
is the same as saying
that Dim(
E
T
,α
)
>
1.
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 Fall '07
 JURY
 Calculus, Matrices, Diagonalizable matrix, Singular value decomposition

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