Solution to problem 6
Due Tuesday November 8
Rui Zhang and T. Witten 11/12/05
properties of determinants
A generalized determinant of a set of
n
arbitrary vectors
X
1
, ...,
X
n
in
an
n
dimensional space is defined as a linear realvalued function
f
(
X
1
,
X
2
, ...,
X
n
) in each of the
n
vectors that changes sign when any adjacent pair
X
i
,
X
i
+1
are interchanged. The ordinary determinant
is a specific case of a generalized determinant. If we have a basis
e
1
, ...,
e
n
, we can express
f
(
X
...
)
in terms of the
f
(
e
...
) and the set of coefficients
x
ij
in
X
i
=
∑
n
j
=1
x
ij
e
j
.
Thus the
f
of the vectors
becomes a function on the matrix of
x
ij
. For a given set of
X
s
we define the corresponding matrix
as X
¯
. This problem is a review of standard properties of determinants. Feel free to consult standard
sources in writing out the answers to these questions.
a) Is it possible to make a sequences of adjacent interchanges that return to the original ordering
yet that result in a net change of sign? Why not?
Solution
: Of course it’s not possible, otherwise the function
f
becomes multivalued, and is not
defined properly. Now we need to show that, if there are
N
adjacent interchanges before the original
ordering is returned,
N
must be an even number. For
n
different real numbers
x
1
, x
2
, ...x
n
, we consider
the polynomial
P
(
x
1
, x
2
, ...x
n
) =
i<j
(
x
i

x
j
)
.
(1)
Clearly
P
is nonzero for any set of distinct
x
’s. We show below that it changes sign under all adjacent
interchanges. We consider a sequence of
N
interchanges that restores the original order. If
N
were
odd, we would have
P
(
x
1
, ..., x
n
) =

P
(
x
1
, ..., x
n
), so that
P
(
x
1
, ..., x
n
) = 0. Since
P
is manifestly
not 0, we conclude that
N
cannot be odd. If
x
i
and
x
i
+1
are exchanged, the affected factors in
P
are (
x
k

x
i
)(
x
k

x
i
+1
) with
k < i
, (
x
i

x
k
)(
x
i
+1

x
k
) with
k > i
+ 1 and (
x
i

x
i
+1
). Only the
(
x
i

x
i
+1
) factor changes sign; the others are unchanged. Thus
P
changes sign.
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 Spring '10
 Pucker
 Linear Algebra, mechanics, Determinant, Vector Space, basis, f Xσ, f Yσ

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