A Determinantal Identity - Nonlinear Oligopolies: Stability and Bifurcations - page 306

Appendix E
A Determinantal Identity

In analysing the local asymptotic stability of discrete dynamic oligopolies the eigen-
value equation of the associated Jacobians have to be determined. The Jacobians
have similar special structure which allows us to give a simple representation of
their characteristic polynomials.
Our method is based on the following simple identity.
Lemma E.1. Let a ; b ; 2 R

N

be two real column vectors, then

T / D 1 C ab T :

det . I C ab (E.1)

Proof. Let D N denote this determinant. We will use finite induction with respect to
N to prove identity (E.1). If N D 1,then

D det .1 C a 1 b 1 / D 1 C a 1 b 1

D 1

so (E.1) clearly holds. If N>1, then with the notation a D .a i / and b D b i we
have

0
@

1
A

1 C a 1 b 1 a 1 b 2 ::: a 1 b N
a 2 b 1 1 C a 2 b 2 ::: a 2 b N
: : :

D N D det :

: : : : : :

a N b 1 a N b 2 :::1 C a N b N

Subtract the a N =a N 1 -multiple of row N 1 from the last row, then the a N 1 =a N 2
-multiple of row N 2 from row N 1, and so on, and finally subtract the a 2 =a 1 -
multiple of row 2 from the first row. Then the value of the determinant remains the
same, so

0 1

1 C a 1 b 1 a 1 b 2 ::: a 1 b N 1 a 1 b N

@ A

a a 1

1

a a 2

D N D det ;

: : :

1

a N
a N 1

1

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