Unitary Matrices - Texas A&M University

1 Chapter 4. Unitary Matrices 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 k k2.

2 We begin by recalling a few important facts. Recall that a set of vectors x1 , .. , xk Cn is called orthogonal if x j xm= hxm , xj i = 0 for 1 j 6= m k. The set is orthonormal if . 1 j=m xj xm = mj =. 0 j 6= m. An orthogonal set of vectors can be made orthonormal by scaling: 1. xj xj . (x j xj )1/2. Theorem Every set of orthonormal vectors is linearly independent. Proof. The proof is routine, using a common technique. Suppose S =. {xj }kj=1 is orthonormal and linearly dependent. Then, without loss of gen- 157. 158 CHAPTER 4. Unitary Matrices . erality (by relabeling if needed), we can assume k 1. X. uk = cj uj . j=1. Compute u k uk as . Xk 1. 1 = u k uk = u k cj uj . j=1. k 1. X. = cj u k uj = 0. j=1. This contradiction proves the result.

3 Corollary If S = {u1 .. uk } Cn is orthonormal then k n. Corollary Every k-dimensional subspace of Cn has an orthonormal basis. Proof. Apply the Gram Schmidt process to any basis to orthonormalize it. Definition A matrix U Mn is said to be Unitary if U U = I. [If U Mn (R) and U T U = I, then U is called real orthogonal.]. Note: A linear transformation T : Cn Cn is called an isometry if kT xk = kxk for all x Cn . Proposition Suppose that U Mn is Unitary . (i) Then the columns of U form an orthonormal basis of Cn , or Rn , if U is real. (ii) The spectrum (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.

4 , n. P If is an Peigenvalue of U with pertaining eigenvector x, then kU xk = k xi ui k = ( |xi |2 )1/2 = kxk Q =. | |kxk. Hence | |. Q = 1. Finally, (iii) follows directly because det U = i . Thus | det U | = | i | = 1. This important result is just one of many equivalent results about Unitary Matrices . In the result below, a number of equivalences are established. Theorem Let U Mn . The following are equivalent. BASICS 159. (a) U is Unitary . (b) U is nonsingular and U = U 1 . (c) U U = 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.

5 (b) (c) follows from the fact that a left inverse is also a right inverse. (a) (d) U U = (U ) U = I. (d) (e) (e) (b) u j uk jk , where u1 .. 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 , we have n X. Ux = xi ui 1. where x = (x1 , .. , xn )T . It is an easy matter to see that n X. kU xk2 = |xi |2 = kxk2 . 1. (g) (e). Consider x = ej . Then U x = uj . Hence 1 = kej k = kU ej k =. kuj k. The columns of U have norm one. Now let x = ei + ej be chosen 160 CHAPTER 4. Unitary Matrices . q such that kxk = k ei + ej k = | |2 + | |2 = 1. Then 1 = kU xk2. = kU ( ei + ej )k2. = hU ( ei + ej ) , U ( ei + ej )i = | |2 hU ei , U ei i + | |2 hU ej , U ej i + hU ei , U ej i + hU ej , U ei i = | |2 hui , ui i + | |2 huj , uj i + hui , uj i + huj , ui i.

6 = | |2 + | |2 + 2< hui , uj i . = 1 + 2< hui , uj i . Thus < hui , uj i = 0. Now suppose hui , uj i = s + it. Selecting =. = 12 we obtain that < hui , uj i = 0, and selecting = i = 12 we obtain that = hui , uj i = 0. Thus hui , uj i = 0. Since the coordinates i and j are arbitrary, it follows that the columns of U are orthogonal. (g) (h) Suppose {v1 , .. , vn } is orthogonal. For any two of them kU (vj +. vk )k2 = kvj + vk k2 . Hence hU vj , U vk i = 0. (h) (e) The orthormal set of the standard unit vectors ej , j = 1, .. , n is carried to the columns of U . That is U ej = uj , the j th column of U . Therefore the columns of U are orthonormal. Corollary If U Mn (C) is Unitary , then the transformation defined by U preserves angles.

7 Proof. We have for any vectors x, y Cn that the angle is completely hx,yi determined from the inner product via cos = kxkkyk . Since U is Unitary (and thus an isometry) it follows that hU x, U yi = hU U x, yi = hx, yi This proves the result. cos sin . Example Let T ( ) = sin cos . where is any real. Then T ( ) is real orthogonal. Proposition If U M2 (R) is real orthogonal, then U has the form T ( ) for some or the form . 1 0 cos sin . U= T ( ) =. 0 1 sin cos . Finally, we can easily establish the diagonalizability of Unitary Matrices . BASICS 161. Theorem If U Mn is Unitary , then it is diagonalizable. Proof. To prove this we need to revisit the proof of Theorem As before, select the first vector to be a normalized eigenvector u1 pertaining to 1.

8 Now choose the remaining vectors to be orthonormal to u1 . This makes the matrix P1 with all these vectors as columns a Unitary matrix. Therefore B1 = P 1 U P is also Unitary . However it has the form . 1 .. n 1. 0 .. B1 = . where A2 is (n 1) (n 1).. A2 . 0. Since it is Unitary , it must have orthogonal columns by Theorem It follows then that 1 = 2 = = n = 0 and . 0 .. 0. 0 .. B1 = .. 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 Let U Mn be Unitary . Then (i) Then U has a set of n orthogonal eigenvectors. (ii) Let { 1.}

9 , n } 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 X. U= j vj vjT. j=1. This representation is often called the spectral respresentation or spectral decomposition of U. 162 CHAPTER 4. Unitary Matrices . 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 A group G is a set with a binary operation G G G.

10 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, then ab G. 2. Associativity. If a, b, c G, then a(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 element a G such that aa = a a = e. a is called the inverse of a and is often denoted by a 1 . 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.