7.3 Distance and Unitary Matrices
Some matrices preserve lengths of vectors. That is
. Such a matrix is called
unitary. 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 Problem 19
on Page 310
. The traditional definition
of unitary is as follows.
Definition 7.3.1 Let U ∈ Mn×n. Then U is called unitary if U∗U = UU∗ = I. When U consists
entirely of real entries, a unitary matrix is called an orthogonal matrix.
Then the following proposition relates this to preservation of lengths of vectors.
Proposition 7.3.2 An n × n matrix U is unitary 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 ℂn
and let θ ∈ ℂ
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 = U∗Uu − u. It follows that
This is true for all u and so U∗U = I. Thus it is also true that UU∗ = I. One can use the fact shown in
Problem 19 on Page 310.
Conversely, if U∗U = I, then
Thus U preserves distance. ■