Engineering Math

7.3 Distance and Unitary Matrices

Some matrices preserve lengths of vectors. That is

= 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.

Then the following proposition relates this to preservation of lengths of vectors.

Proof: First suppose the matrix U preserves all lengths. Since U preserves distances,

= for every u. Let u,v be arbitrary vectors in ℂⁿ and let θ ∈ ℂ, = 1, and θ = . Therefore from the axioms of the inner product,

= (U θu,U θu)+ (Uv,U v)+ (Uθu,U v)+ (Uv,U θu)

and so, subtracting the ends, it follows that for all u,v,

0 = 2 Reθ (U ∗Uu − u,v) = 2 |(U∗U u− u,v )|

from the above choice of θ. Now let v = U^∗Uu − u. It follows that

U ∗Uu − u = (U ∗U − I)u = 0.

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

|U u|2 = (U u,Uu) = (U ∗Uu,u ) = (u,u) = |u|2

Thus U preserves distance. ■