Serge Ballif
MATH 535 Homework 5
September 28, 2007
1. a) Let
A
= (
a
ij
)
∈
Mat
n
×
m
(
F
)
, such that for each
i
,
∑
n
j
=1
a
ij
= 1
. Prove that
1
is an eigen
value of
A
.
b) Show that, if instead
∑
n
i
=1
a
ij
= 1
for
j
= 1
,...,n
, then
1
is still an eigenvalue of
A
.
a) The vector of 1’s
v
=
±
1
.
.
.
1
²
is an eigenvector for the eigenvalue
λ
= 1, since
A
±
1
.
.
.
1
²
=
∑
n
j
=1
a
1
,j
.
.
.
∑
n
j
=1
a
n,j
!
= 1
·
v
.
b) In this case we know that
A
t
satisﬁes the identity in part a). Hence,
A
t
has 1
as an eigenvalue. We also know that det
A
= det
A
t
, so
χ
A
=
χ
A
t
. Thus,
A
and
A
t
have the same eigenvalues. Therefore
A
has eigenvalue 1.
2. Show that there are no proper twosided ideals
I
in the ring
Mat
n
×
n
(
F
)
when
F
is a ﬁeld. Hint:
Show that if
A
∈
I
⊂
Mat
n
×
n
(
F
)
, then elementary row and column operations can be used to
show that
I
n
∈
I
.
Let
I
⊂
Mat
n
×
n
(
F
) be a nonzero ideal. Let
K
= (
k
ij
) be a nonzero element of
I
. Denote by
E
ij
the matrix with 1 in the
i,j
th slot, and 0 elsewhere. Then
the product
E
ab
(
k
ij
)
E
cd
is the matrix with
k
bc
in the
a,d
th slot and 0 elsewhere.
Thus,
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 Fall '08
 BROWNAWELL,WOODRO
 Math, Algebra, Matrices, Diagonal matrix, Triangular matrix, KBC, nonzero diagonal entries

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