Chapter 4 Unitary Matrices 4.1 Basics This chapter considers a very important class of matrices that are quite use- ful in proving a number of structure theorems about all matrices. Called unitary matrices, they comprise a class of matrices that have the remarkable properties that as transformations they preserve length, and preserve the an- gle between vectors. This is of course true for the identity transformation. Therefore it is helpful to regard unitary matrices as “generalized identities,” though we will see that they form quite a large class. An important exam- ple of these matrices, the rotations, have already been considered. In this chapter, the underlying field is usually C, the underlying vector space is Cn, and almost without exception the underlying norm is 2.Webeginby recalling a few important facts. k·k Recall that a set of vectors x1,...,xk Cn is called orthogonal if ∈ x xm = xm,xj =0for1 j = m k.Thesetisorthonormal if j∗ h i ≤ 6 ≤ 1 j = m x x = δ = j∗ m mj 0 j = m. ½ 6 An orthogonal set of vectors can be made orthonormal by scaling: 1 x x . j 1/2 j −→ (xj∗xj) Theorem 4.1.1. Every set of orthonormal vectors is linearly independent. Proof. The proof is routine, using a common technique. Suppose S = k xj is orthonormal and linearly dependent. Then, without loss of gen- { }j=1 157 158 CHAPTER 4. UNITARY MATRICES erality (by relabeling if needed), we can assume k 1 − uk = cjuj. j=1 X Compute uk∗uk as k 1 − 1=u∗uk = u∗ cjuj k k j=1 X k 1 − = cjuk∗uj =0. j=1 X This contradiction proves the result. Corollary 4.1.1. If S = u1 ...uk Cn is orthonormal then k n. { } ⊂ ≤ Corollary 4.1.2. Every k-dimensional subspace of Cn has an orthonormal basis. Proof. Apply the Gram—Schmidt process to any basis to orthonormalize it. Definition 4.1.1. AmatrixU Mn is said to be unitary if U ∗U = I.[If T ∈ U Mn(R)andU U = I,thenU is called real orthogonal.] ∈ Note: A linear transformation T : Cn Cn is called an isometry if → Tx = x for all x Cn. k k k k ∈ Proposition 4.1.1. Suppose that U Mn is unitary. (i) Then the columns ∈ of U form an orthonormal basis of Cn,orRn ,ifU is real. (ii) The spectrum σ(u) z z =1 . (iii) det U =1. ⊂ { || | } | | Proof. The proof of (i) is a consequence of the definition. To prove (ii), first denote the columns of U by ui,i=1,...,n.Ifλ is an eigenvalue of U with 2 1/2 pertaining eigenvector x,then Ux = xiui =( xi ) = x = k k k k | | k k λ x .Henceλ = 1. Finally, (iii) follows directly because det U = λi. | |k k | | P P Thus det U = λi =1. | | | | Q This importantQ result is just one of many equivalent results about unitary matrices. In the result below, a number of equivalences are established. Theorem 4.1.2. Let U Mn. The following are equivalent. ∈ 4.1. BASICS 159 (a) U is unitary. 1 (b) U is nonsingular and U ∗ = U − . (c) UU∗ = I. (d) U ∗ is unitary. (e) The columns of U form an orthonormal set. (f) The rows of U form an orthonormal set. (g) U is an isometry. (h) U carries every set of orthonormal vectors to a set of orthonormal vectors. Proof. (a) (b) follows from the definition of unitary and the fact that the inverse⇒ is unique. (b) (c) follows from the fact that a left inverse is also a right inverse. ⇒ (a) (d) UU =(U ) U = I. ⇒ ∗ ∗ ∗ ∗ (d) (e) (e) (b) uj∗ukδjk,whereu1 ...un are the columns of U. Similarly⇒ (b) ≡(e). ⇒ ⇒ (d) (f) same reasoning. ≡ (e) (g) We know the columns of U are orthonormal. Denoting the columns ⇒ by u1 ...un,wehave n Ux = xiui 1 X T where x =(x1,... ,xn) . It is an easy matter to see that n 2 2 2 Ux = xi = x . k k | | k k 1 X (g) (e). Consider x = ej.ThenUx = uj.Hence1=ej = Uej = ⇒ k k k k uj .ThecolumnsofU have norm one. Now let x = αei + βej be chosen k k 160 CHAPTER 4. UNITARY MATRICES 2 2 such that x = αei + βej = α + β =1. Then k k k k | | | | q 1= Ux 2 k k 2 = U (αei + βej) k k = U (αei + βej) ,U(αei + βej) h 2 2 i = α Uei,Uei + β Uej,Uej + αβ¯ Uei,Uej +¯αβ Uej,Uei | |2 h i 2| | h i h i h i = α ui,ui + β uj,uj + αβ¯ ui,uj +¯αβ uj,ui | |2 h 2i | | h i h i h i = α + β +2 αβ¯ ui,uj | | | | < h i =1+2 αβ¯ ui,uj < h ¡ i ¢ Thus αβ¯ ui,u¡ j =0. ¢Now suppose ui,uj = s + it. Selecting α = 1< h i h i 1 β = we obtain that ui,uj = 0, and selecting α = iβ = we obtain √2 ¡ ¢ <h i √2 that ui,uj =0. Thus ui,uj = 0. Since the coordinates i and j are arbitrary,=h it followsi that theh columnsi of U are orthogonal. (g) (h) Suppose v1,...,vn is orthogonal. For any two of them U(vj + ⇒2 2 { } k vk) = vj + vk . Hence Uvj,Uvk =0. k k k h i (h) (e) The orthormal set of the standard unit vectors ej,j=1,...,n ⇒ th is carried to the columns of U.ThatisUej = uj,thej column of U. Therefore the columns of U are orthonormal. Corollary 4.1.3. If U Mn(C) is unitary, then the transformation defined by U preserves angles. ∈ Proof. We have for any vectors x, y Cn that the angle θ is completely ∈ x,y determined from the inner product via cos θ = hx yi .SinceU is unitary (and thus an isometry) it follows that k kk k Ux,Uy = U ∗Ux,y = x, y h i h i h i This proves the result. cos θ sin θ Example 4.1.1. Let T (θ)= sin θ −cos θ where θ is any real. Then T (θ)is real orthogonal. £ ¤ Proposition 4.1.1. If U M2(R) is real orthogonal, then U has the form T (θ) for some θ or the form∈ 10 cos θ sin θ U = T (θ)= 0 1 sin θ cos θ · − ¸ · − ¸ Finally, we can easily establish the diagonalizability of unitary matrices. 4.1. BASICS 161 Theorem 4.1.3. If U Mn is unitary, then it is diagonalizable. ∈ Proof. To prove this we need to revisit the proof of Theorem 3.5.2. As before, select the first vector to be a normalized eigenvector u1 pertaining to λ1. Now choose the remaining vectors to be orthonormal to u1.This makes the matrix P1 with all these vectors as columns a unitary matrix. 1 Therefore B1 = P − UP is also unitary. However it has the form λα1 ... αn 1 0 − B = where A is (n 1) (n 1) 1 . 2 . A2 − × − 0 Since it is unitary, it must have orthogonal columns by Theorem 4.1.2. It follows then that α1 = α2 = = αn =0 and ··· λ 0 ... 0 0 B = 1 . A2 0 At this point one may apply and inductive hypothesis to conclude that A2 is similar to a diagonal matrix. Thus by the manner in which the full similarity was constructed, we see that A must also be similar to a diagonal matrix. Corollary 4.1.1. Let U Mn be unitary. Then ∈ (i) Then U has a set of n orthogonal eigenvectors. (ii) Let λ1,...,λn and v1,...,vn denote respectively the eigenvalues and{ their pertaining} orthonormal{ } eigenvectors of U. Then U has the representation as the sum of rank one matrices given by n T U = λjvjvj j=1 X This representation is often called the spectral respresentation or spectral decomposition of U. 162 CHAPTER 4. UNITARY MATRICES 4.1.1 Groups of matrices Invertible and unitary matrices have a fundamental structure that makes possible a great many general statements about their nature and the way they act upon vectors other vectors matrices. A group is a set with a math- ematical operation, product, that obeys some minimal set of properties so as to resemble the nonzero numbers under multiplication. Definition 4.1.2. A group G isasetwithabinaryoperationG G G which assigns to every pair a, b of elements of G a unique element×ab in→G. The operation, called the product,satisfies four properties: 1. Closure. If a, b G,thenab G. ∈ ∈ 2. Associativity. If a, b, c G,thena(bc)=(ab)c. ∈ 3. Identity. There exists an element e G such that ae = ea = a for every a G. e is called the identity of∈ G. ∈ 4. Inverse. For each a G, there exists an elementa ˆ G such that ∈ ∈ 1 aaˆ =ˆaa = e.ˆa is called the inverse of a and is often denoted by a− . A subset of G that is itself a group under the same product is called a subgroup of G. It may be interesting to note that removal of any of the properties 2-4 leads to other categories of sets that have interest, and in fact applications, in their own right. Moreover, many groups have additional properties such as commutativity, i.e. ab = ba for all a, b G.Belowareafewexamplesof matrix groups. Note matrix addition is not∈ involved in these definitions. Example 4.1.2. As usual Mn is the vector space of n n matrices.