19.8. DIAGONALIZATION OF SYMMETRIC MATRICES 421

= (Uθu,Uθu)+(Uv,Uv)+(Uθu,Uv)+(Uv,Uθu)

= |θu|2 + |v|2 +θ(UTUu,v

)+θ

(v,UTUu

)= |u|2 + |v|2 +2θ

(UTUu,v

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

0 = 2θ(UTUu−u,v

)= 2

∣∣(UTUu−u,v)∣∣

from the above choice of θ . Now let v =UTUu−u. It follows that

UTUu−u=(UTU − I

)u= 0.

This is true for all u and so UTU = I. Thus it is also true that UUT = I.Conversely, if UTU = I, then

|Uu|2 = (Uu,Uu) =(UTUu,u

)= (u,u) = |u|2

Thus U preserves distance.

19.8 Diagonalization of Symmetric MatricesRecall that a symmetric matrix is a real n× n matrix A such that AT = A. One nice thingabout symmetric matrices is that they have only real eigenvalues. You might want to reviewthe property of the conjugate which says that zw = z̄w̄ and how the conjugate of a sum isthe sum of the conjugates.

Proposition 19.8.1 Suppose A is a real symmetric matrix. Then all eigenvalues arereal.

Proof: Suppose Ax= λx. Then

xT Ax= xTλx= λxTx= λxTx

The last step happens because both xTx and xTx are the sum of the entries of x times theconjugate of these entries. Also xT Ax is some complex number, a 1× 1 matrix and so itequals its transpose. Thus, since A = AT ,

xT Ax= xT ATx= xT Ax= xT Ax= xTλx= λxTx

Since x ̸= 0, xTx is a positive real number. Hence, the above shows that λ = λ .

Definition 19.8.2 A set of vectors in Rp {x1, · · · ,xk} is called an orthonormal setof vectors if

xTi x j = δ i j ≡

{1 if i = j0 if i ̸= j

Note this is the same as saying that (xi,x j) = xi ·x j = δ i j.