332 CHAPTER 13. MATRICES AND THE INNER PRODUCT

The equation, U∗AU = D implies

AU =(

Au1 Au2 · · · Aun

)= UD =

(λ 1u1 λ 2u2 · · · λ nun

)where the entries denote the columns of AU and UD respectively. Therefore, Aui = λ iuiand since the matrix is unitary, the i jth entry of U∗U equals δ i j and so

δ i j = uTi u j = uT

i u j = ui ·u j.

This proves the corollary because it shows the vectors {ui} form an orthonormal basis.In case A is real and symmetric, simply ignore all complex conjugations in the aboveargument. ■

This theorem is particularly nice because the diagonal entries are all real. What of amatrix which is unitarily similar to a diagonal matrix without assuming the diagonal entriesare real? That is, A is an n×n matrix with

U∗AU = D

Then this requiresU∗A∗U = D∗

and so since the two diagonal matrices commute,

AA∗ = UDU∗UD∗U∗ =UDD∗U∗ =UD∗DU∗

= UD∗U∗UDU∗ = A∗A

The following definition describes these matrices.

Definition 13.1.7 An n×n matrix is normal means: A∗A = AA∗.

We just showed that if A is unitarily similar to a diagonal matrix, then it is normal. Theconverse is also true. This involves the following lemma.

Lemma 13.1.8 If T is upper triangular and normal, then T is a diagonal matrix. If A isnormal and U is unitary, then U∗AU is also normal.

Proof: This is obviously true if T is 1× 1. In fact, it can’t help being diagonal in thiscase. Suppose then that the lemma is true for (n−1)× (n−1) matrices and let T be anupper triangular normal n×n matrix. Thus T is of the form

T =

(t11 a∗

0 T1

), T ∗ =

(t11 0T

a T ∗1

)Then

T T ∗ =

(t11 a∗

0 T1

)(t11 0T

a T ∗1

)=

(|t11|2 +a∗a a∗T ∗1

T1a T1T ∗1

)

T ∗T =

(t11 0T

a T ∗1

)(t11 a∗

0 T1

)=

(|t11|2 t11a

∗

at11 aa∗+T ∗1 T1

)