19.7 Distance and Orthogonal Matrices
Some matrices preserve lengths of vectors. That is
Such a matrix is called orthogonal. Actually, this is not the standard definition.
The standard definition is given next. First recall that if you have two square
matrices of the same size and one acts like the inverse of the other on one side,
then it will act like the inverse on the other side as well. See, for example, the
discussion after Theorem 18.10.12
. The traditional definition of orthogonal is as
Definition 19.7.1 Let U be a real n × n matrix. Then U is called
orthogonal if UTU = UUT = I.
Then the following proposition relates this to preservation of lengths of vectors.
Proposition 19.7.2 An n×n matrix U is orthogonal if and only if
for all vectors x.
Proof: First suppose the matrix U preserves all lengths. Since U preserves distances,
be arbitrary vectors in ℝn
and let θ ∈ ℝ
. Therefore from the axioms of the inner product
and so, subtracting the ends, it follows that for all u,v,
from the above choice of θ. Now let v = UTUu − u. It follows that
This is true for all u and so UTU = I. Thus it is also true that UUT = I.
Conversely, if UTU = I, then
Thus U preserves distance. ■