Page 308 - Advanced Linear Algebra
P. 308
292 Advanced Linear Algebra
!
8
´µ 4´µ ~ 4 8
8
8
Taking determinants gives
det²4 ³ ~ det²´ µ ³ det²4 ³
8
8
8
Therefore, if is nonsingular, then
=
det²´ µ ³ ~ f
8
Since the determinant is an invariant under similarity, we have the following
theorem.
Theorem 11.31 Let be an orthogonal transformation on a nonsingular
orthogonal geometry .
=
=
1 )det²´ µ ³ is the same for all ordered bases for and
8 8
det²´ µ ³ ~ f
8
This determinant is called the determinant of and denoted by det²³ .
)
2 If det²³ ~ , then is called a rotation and if det²³ ~ c , then is
called a reflection .
3 The set E ) b of rotations is a subgroup of the orthogonal group E ²= ³ ²= ³
and the determinant map det¢²= ³ ¦ ¸c Á ¹ is an epimorphism with
E
kernel E b . Hence, if char ²-³ £ , then E ²= ³ b ²= ³ is a normal subgroup
of E²= ³ of index .
Symmetries
is defined as an
Recall again that for a real inner product space, a reflection / "
operator for which
/" ~ c"Á /$ ~ $ for all $ º"»
"
"
and that
º%Á "»
/% ~ % c "
"
º"Á "»
In particular, if char²-³ £ and " = is nonisotropic, then span²"³ is
nonsingular and so
= ~ span ²"³ p span ²"³
is well-defined and, in the context of general orthogonal
Then the reflection / "
geometries, is called the symmetry determined by and we will denote it by
"
" " . We can also write ~c p , that is,
" ²% b&³ ~ c% b&
for all % span ²"³ and & span ²"³ .
