11.4. SCHUR’S THEOREM 191

is given next. First recall that if you have two square matrices of the same size and one actslike the inverse of the other on one side, then it will act like the inverse on the other side aswell. See Problem 19 on Page 168. The traditional definition of unitary is as follows.

Definition 11.3.1 Let U ∈ Mn×n. Then U is called unitary if U∗U = UU∗ = I. When Uconsists 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 11.3.2 An 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,|Uu| = |u| for every u. Let u,v be arbitrary vectors in Cn and let θ ∈ C, |θ | = 1, andθ (U∗Uu−u,v) = |(U∗Uu−u,v)|. Therefore from the axioms of the inner product,

|u|2 + |v|2 +2Reθ (u,v) = |θu|2 + |v|2 +θ (u,v)+ θ̄ (v,u)

= |θu+v|2 = (U (θu+v) ,U (θu+v))

= (Uθu,Uθu)+(Uv,Uv)+(Uθu,Uv)+(Uv,Uθu)

= |θu|2 + |v|2 +θ (U∗Uu,v)+ θ̄ (v,U∗Uu)

= |u|2 + |v|2 +2Reθ (U∗Uu,v)

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

0 = 2Reθ (U∗Uu−u,v) = 2 |(U∗Uu−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 thefact shown in Problem 19 on Page 168.

Conversely, if U∗U = I, then

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

Thus U preserves distance. ■

11.4 Schur’s TheoremThe most significant theorem about eigenvalues and eigenvectors in the space of n×n com-plex matrices is Schur’s theorem. First is a simple version of the Gram Schmidt theorem.

Definition 11.4.1 A set of vectors in Fn,F = R or C, {x1, · · · ,xk} is called an orthonor-mal set of vectors if

xTi x j = x∗i x j = δ i j ≡

{1 if i = j0 if i ̸= j