Some matrices preserve lengths of vectors. That is
|Ux|
=
|x|
for any x in ℂn. 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 7.3.1Let U ∈ Mn×n. Then U is called unitary if U∗U = UU∗ = I. When U consistsentirely of real entries, a unitary matrix is called an orthogonal matrix.
Then the following proposition relates this to preservation of lengths of vectors.
Proposition 7.3.2An n × n matrix U is unitary if and only if
|Ux|
=
|x |
for all vectors x.
Proof:First suppose the matrix U preserves all lengths. Since U preserves distances,
|U u|
=
|u|
for
every u. Let u,v be arbitrary vectors in ℂn and let θ ∈ ℂ,