Page 196 - Advanced Linear Algebra

180 Advanced Linear Algebra

When ~ , we can write ²%³ as

²%Â Á ³ ~ b % b % ~ %²% b ³ b

which looks suspiciously like a determinant:
%
²%Â Á ³ ~ det > ?

c % b
c
~ % det 6 0 c > ? 7
c
~ det ²%0 c *´ ²%³µ³

So, let us define
³ ~ %0 c *´ ²%³µ
(²%Â Á Ã Á c
v % Ä y
x c % Ä {
~ x x c Æ Å { {
x {
Å Å Æ % c
w Ä c % b c z
where is an independent variable. The determinant of this matrix is a
%
%
polynomial in whose degree equals the number of parameters ÁÃÁ c .
We have just seen that
det²(²%Â Á ³³ ~ ²%Â Á ³

and this is also true for ~ . As a basis for induction, if
³
det²(²%Â ÁÃÁ c ³³ ~ ²%Â ÁÃÁ c

then expanding along the first row gives
det²(²%Á ÁÃÁ ³³

v c % Ä y
x c Æ {
~ % det²(²%Á ÁÃÁ ³³ b ²c ³ detx {

Å Å Æ %
w Ä c z
d
~ % det²(²%Á ÁÃÁ ³³ b

~% ²%Â Á Ã Á ³ b

~ % b % b Ä b % b% b b

²%Â ÁÃÁ ³
~ b
We have proved the following.