1
Math 1b Practical
Isometries
March 4, 2009
We have explained that a linear transformation of
R
n
to itself is an isometry when
its matrix is an orthogonal matrix. [The converse is also true.]
An example of an isometry is the reﬂection through a subspace
U
of
R
n
.
The
reﬂection of
x
through
U
is defined as follows: if
x
=
u
+
w
with
u
∈
U
and
w
∈
U
⊥
,
then the reﬂection reﬂ
U
(
x
) is simply
u
−
w
. It follows that
reﬂ
U
(
x
) = 2 proj
U
(
x
)
−
x
.
The proofs of the following two theorems are short, but understanding and reading
them will be diﬃcult. We will not do all details in class.
Theorem 1.
Every
n
by
n
orthogonal matrix
Q
is the product of at most
n
matrices of
reflections through hyperplanes.
Proof:
Recall that a hyperplane
H
is an (
n
−
1)dimensional subspace. We will use
R
H
to denote the matrix [reﬂ
H
] of the reﬂection through
H
. So
R
H
is an orthogonal matrix
and
R
2
H
=
I
.
Let
U
=
{
x
:
Q
x
=
x
}
be the eigenspace of
Q
corresponding to eigenvalue 1, i.e.
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 Winter '09
 Wilson
 Math, Linear Algebra, Algebra, Matrices, Singular value decomposition, Diagonal matrix, Orthogonal matrix, RH Qz

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