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Math 1b Practical — February 22, 2011
Digraphs, nonnegative matrices, Google page ranking
— Nonnegative matrices —
Theorem 1.
Let
A
be a square matrix of nonnegative real numbers. Then
A
has an
eigenvector
e
all of whose entries are nonnegative.
Proof:
Omitted.
ut
There may be several linearly independent nonnegative eigenvectors. For example, the
identity matrix
I
has many nonnegative eigenvectors (all corresponding to eigenvalue 1).
A nonnegative diagonal matrix with distinct diagonal entries has nonnegative eigenvectors
corresponding to diferent eigenvalues.
Here is a stronger version o± Theorem 1, and also a strengthening when the nonnegative
matrix is
irreducible
in a sense to be de²ned later.
Theorem 2.
Let
A
be a square matrix of nonnegative real numbers. Let
λ
be the max
imum of the absolute values

μ

of the eigenvalues
μ
of
A
.T
h
e
n
λ
is an eigenvalue of
A
and there are nonnegative eigenvectors coresponding to
λ
.
Also omitted.
ut
Theorem 3 (PerronFrobenius).
Let
A
be an irreducible square nonnegative matrix.
Then
A
has a unique nonnegative eigenvector
e
, up to scalar multiples. This eigenvector
e
has all positive entries and the corresponding eigenvalue
λ
has algebraic multiplicity
1
and the property that
λ
≥
μ

for every eigenvalue
μ
of
A
.
Omitted yet again, though we will prove it ±or probability matrices (to be de²ned
later) below.
The eigenvalue
λ
in Theorem 2 or 3 may be called the
dominant eigenvalue
o± the
matrix
A
. We remark that there may be real or complex eigenvalues
μ
6
=
λ
but so that

μ

=
λ
.
Such eigenvalues, o± course, do not correspond to nonnegative eigenvectors,
because
μ
would be negative or complex. For example,
±
01
10
²
has eigenvalues +1 and
−
1.
(In general, the maximum o± the absolute values o± the eigenvalues o± a matrix is called
the
spectral radius
o± the matrix.)
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Example 1.
The eigenvalues of
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0500000030
0000100014
0040003014
1520035000
2020003005
0000004010
0133000005
2100231542
0000220302
0500040043
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
are
11
.
268
,
5
.
167
,
−
3
.
556
±
2
.
351
i,
3
.
269
,
−
2
.
178
±
2
.
212
i,
2
.
167
±
1
.
734
i,
−
0
.
570
.
For this matrix, there are three nonnegative eigenvalues, but only one nonnegative eigen
vector, up to scalar multiples, namely
(0
.
148
,
0
.
161
,
0
.
343
,
0
.
329
,
0
.
310
,
0
.
142
,
0
.
328
,
0
.
572
,
0
.
286
,
0
.
305)
>
.
Theorem 4.
An eigenvalue
λ
of a nonnegative matrix
A
=(
a
ij
)
that corresponds to a
nonnegative eigenvector is at least the smallest row sum and at least the smallest column
sum of
A
, and is at most the largest row sum and at most the largest column sum of
A
.
That is,
min
i
n
±
j
=1
a
ij
≤
λ
≤
max
i
n
±
j
=1
a
ij
and
min
j
n
±
i
=1
a
ij
≤
λ
≤
max
j
n
±
i
=1
a
ij
.
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 Winter '08
 Aschbacher
 Real Numbers, Matrices, nonnegative eigenvector, nonnegative eigenvectors

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