Page 422 - Advanced Linear Algebra
P. 422
406 Advanced Linear Algebra
Theorem 14.22 If (Á ) 4 ²-³ , then
²()³ ~ ²(³ ²)³
Proof. Consider the map ¢ 4 ²-³ ¦ - defined by
(
²?³ ~ ²(?³
(
We can consider as a function on the columns of ? ( and think of it as a
composition
² ³
² ³
¢ ²? ÁÃÁ? ² ³ ³ ª ²(? Á ÃÁ(? ² ³ ³ ª ²(?³
(
is multilinear. It is also clear that
Each step in this map is multilinear and so (
( ( is antisymmetric and so is a scalar multiple of the determinant function,
say ²?³ ~ ²?³ . Then
(
²(?³ ~ ²?³ ~ ²?³
(
gives ²(³ ~ and so
Setting ?~ 0
²(?³ ~ ²(³ ²?³
as desired.
Theorem 14.23 A matrix ( 4 ²-³ is invertible if and only if ²(³ £ .
Proof. If 7 4 ²-³ is invertible, then 77 c ~ 0 and so
²7³ ²7 c ³ ~
which shows that ²7³ £ and ²7 c ³ ~ ° ²7³ . Conversely, any matrix
( 4 ²-³ is equivalent to a diagonal matrix
(~ 7+8
where and are invertible and is diagonal with 's and 's on the main
+
7
8
diagonal. Hence,
²(³ ~ ²7³ ²+³ ²8³
and so if ²(³ £ , then ²+³ £ , which happens if and only if + ~ 0 ,
whence is invertible.
(
Exercises
1. Show that if ¢> ¦ ? is a linear map and ¢< d = ¦ > is bilinear,
then k ¢ < d= ¦ ? is bilinear.
2. Show that the only map that is both linear and -linear for ( ) is the
zero map.
3. Find an example of a bilinear map ¢= d = ¦ > whose image
im²³ ~ ¸ ²"Á #³ "Á # = ¹ is not a subspace of > .
