SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

All the vector spaces we have studied thus far in the text are real vector spaces since the scalars are real numbers. A complex vector space is one in which the scalars are complex

numbers. Thus, if v1, v2, . . . , vm are vectors in a complex vector space, then a linear com- bination is of the form ϩ ϩ и и и ϩ c1v1 c 2v2 cmvm n where the scalarsc1, c 2, . . . , cm are complex numbers. The complex version of R is the complex vector spaceC n consisting of ordered n-tuples of complex numbers. Thus, a vector in C n has the form ϭ ͑ ϩ ϩ ϩ ͒ v a1 b1i, a 2 b2i, . . . , an bni . It is also convenient to represent vectors in C n by column matrices of the form ϩ a1 b1i a ϩ b i v ϭ 2 . 2 . ΄ . ΅ ϩ an bni

As with Rn , the operations of addition and scalar multiplication in C n are performed com- ponent by component.

EXAMPLE 1 Vector Operations in Cn Let v ϭ ͑1 ϩ 2i, 3 Ϫ i͒ and u ϭ ͑Ϫ2 ϩ i, 4͒ be vectors in the complex vector spaceC 2 , and determine the following vectors. (a)v ϩ u (b)͑2 ϩ i͒v (c) 3v Ϫ ͑5 Ϫ i͒u

Solution (a) In column matrix form, the sum v ϩ u is 1 ϩ 2i Ϫ2 ϩ i Ϫ1 ϩ 3i v ϩ u ϭ ΄ ΅ ϩ ΄ ΅ ϭ ΄ ΅. 3 Ϫ i 4 7 Ϫ i

(b) Since ͑2 ϩ i͒͑1 ϩ 2i͒ ϭ 5i and ͑2 ϩ i͒͑3 Ϫ i͒ ϭ 7 ϩ i, we have ͑2 ϩ i͒v ϭ ͑2 ϩ i͒͑1 ϩ 2i, 3 Ϫ i͒ ϭ ͑5i, 7 ϩ i͒. (c) 3v Ϫ ͑5 Ϫ i͒u ϭ 3͑1 ϩ 2i, 3 Ϫ i͒ Ϫ ͑5 Ϫ i͒͑Ϫ2 ϩ i, 4͒ (c) ϭ ͑3 ϩ 6i, 9 Ϫ 3i͒ Ϫ ͑Ϫ9 ϩ 7i, 20 Ϫ 4i͒ (c) ϭ ͑12 Ϫ i, Ϫ11 ϩ i͒ 456 CHAPTER 8 COMPLEX VECTOR SPACES

Many of the properties ofRn are shared by Cn. For instance, the scalar multiplicative identity is the scalar 1 and the additive identity in Cn is 0 ϭ ͑0, 0, 0, . . . , 0͒. The standard basis for Cn is simply ϭ ͑ ͒ e1 1, 0, 0, . . . , 0 ϭ ͑ ͒ e2 0, 1, 0, . . . , 0 . . . ϭ ͑ ͒ en 0, 0, 0, . . . , 1

which is the standard basis for Rn . Since this basis contains n vectors, it follows that the dimension of Cn is n. Other bases exist; in fact, any linearly independent set of n vectors in Cn can be used, as we demonstrate in Example 2.

EXAMPLE 2 Verifying a Basis Show that v v v { 1 { 2 { 3 S ϭ ͕͑i, 0, 0͒, ͑i, i, 0͒, ͑0, 0, i͖͒ is a basis for C 3.

3 ͕ ͖ Solution Since C has a dimension of 3, the set v1, v2, v3 will be a basis if it is linearly indepen- dent. To check for linear independence, we set a linear combination of the vectors in S equal to 0 as follows. ϩ ϩ ϭ ͑ ͒ c 1v1 c2v2 c3v3 0, 0, 0 ͑ ͒ ϩ ͑ ͒ ϩ ͑ ͒ ϭ ͑ ͒ c1i, 0, 0 c2i, c2i, 0 0, 0, c3i 0, 0, 0 ͑͑ ϩ ͒ ͒ ϭ ͑ ͒ c1 c2 i, c2i, c3i 0, 0, 0

This implies that ͑ ϩ ͒ ϭ c1 c2 i 0 ϭ c 2i 0 ϭ c 3i 0. ϭ ϭ ϭ ͕ ͖ Therefore,c1 c2 c3 0, and we conclude thatv1, v2, v3 is linearly independent.

EXAMPLE 3 Representing a Vector in Cn by a Basis Use the basis S in Example 2 to represent the vector v ϭ ͑2, i, 2 Ϫ i͒. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 457

Solution By writing ϭ ϩ ϩ v c1v1 c2v2 c3v3 ϭ ͑͑ ϩ ͒ ͒ c1 c2 i, c2i, c3i ϭ ͑2, i, 2 Ϫ i͒ we obtain ͑ ϩ ͒ ϭ c1 c2 i 2 ϭ c 2i i ϭ Ϫ c 3i 2 i ϭ which implies that c2 1 and 2 Ϫ i 2 Ϫ i c ϭ ϭϪ1 Ϫ 2i and c ϭ ϭϪ1 Ϫ 2i. 1 i 3 i Therefore, ϭ ͑Ϫ Ϫ ͒ ϩ ϩ ͑Ϫ Ϫ ͒ v 1 2i v1 v2 1 2i v3. Try verifying that this linear combination yields ͑2, i, 2 Ϫ i͒.

Other than Cn, there are several additional examples of complex vector spaces. For instance, the set of m ϫ n complex matrices with matrix addition and scalar multiplication forms a complex vector space. Example 4 describes a complex vector space in which the vectors are functions.

EXAMPLE 4 The Space of Complex-Valued Functions Consider the set S of complex-valued functions of the form ͑ ϭ ͑ ͒ ϩ ͑ ͒ f x) f1 x if 2 x

wheref1 and f2 are real-valued functions of a real variable. The set of complex numbers form the scalars for S and vector addition is defined by ͑ ͒ ϩ ͑ ͒ ϭ ͓ ͑ ͒ ϩ ͑ ͔͒ ϩ ͓ ͒ ϩ ͑ ͔͒ f x g x f1 x if 2 x g1(x i g2 x ϭ ͓ ͑ ͒ ϩ ͑ ͔͒ ϩ ͓ ͑ ͒ ϩ ͑ ͔͒ f1 x g1 x i f 2 x g2 x . It can be shown that S, scalar multiplication, and vector addition form a complex vector space. For instance, to show that S is closed under scalar multiplication, we let c ϭ a ϩ bi be a complex number. Then ͑ ͒ ϭ ͑ ϩ ͓͒ ͑ ͒ ϩ ͑ ͔͒ c f x a bi f1 x if 2 x ϭ ͓ ͑ ͒ Ϫ ͑ ͔͒ ϩ ͓ ͑ ͒ ϩ ͑ ͔͒ af1 x bf 2 x i bf1 x af 2 x

is in S. 458 CHAPTER 8 COMPLEX VECTOR SPACES

The definition of the Euclidean inner product in Cn is similar to that of the standard dot product in Rn, except that here the second factor in each term is a complex conjugate.

Definition of Let u and v be vectors in Cn. The Euclidean inner product of u and v is given by Euclidean Inner и ϭ ϩ ϩ и и и ϩ u v u1v1 u2v2 unvn. Product in C n

REMARK: Note that if u and v happen to be “real,” then this definition agrees with the standard inner (or dot) product in R n .

EXAMPLE 5 Finding the Euclidean Inner Product in C3 Determine the Euclidean inner product of the vectors u ϭ ͑2 ϩ i, 0, 4 Ϫ 5i͒ and v ϭ ͑1 ϩ i, 2 ϩ i, 0͒.

Solution и ϭ ϩ ϩ u v u1v1 u2v2 u 3v3 ϭ ͑2 ϩ i͒͑1 Ϫ i͒ ϩ 0͑2 Ϫ i͒ ϩ ͑4 Ϫ 5i͒͑0͒ ϭ 3 Ϫ i

Several properties of the Euclidean inner productCn are stated in the following theorem.

Theorem 8.7 Let u, v, and w be vectors in C n and let k be a complex number. Then the following Properties of the properties are true. 1. u и v ϭ v и u Euclidean Inner 2. ͑u ϩ v͒ и w ϭ u и w ϩ v и w Product 3. ͑ku͒ и v ϭ k͑u и v͒ 4. u и ͑kv͒ ϭ k͑u и v͒ 5. u и u Ն 0 6. u и u ϭ 0if and only if u ϭ 0.

Proof We prove the first property and leave the proofs of the remaining properties to you. Let ϭ ͑ ͒ ϭ ͑ ͒ u u1, u 2, . . . , un and v v1, v2, . . . , vn . Then и ϭ ϩ ϩ . . . ϩ v u v1u1 v2u2 vnun ϭ ϩ ϩ . . . ϩ v1u1 v2u 2 vnun ϭ ϩ ϩ . . . ϩ v1u1 v2u 2 vnun SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 459

ϭ ϩ ϩ и и и ϩ u1v1 u2v2 unvn ϭ uи v.

We now use the Euclidean inner product in Cn to define the Euclidean norm (or length) of a vector in Cn and the Euclidean distance between two vectors in Cn .

Definition of Norm The Euclidean norm (or length) of u in Cn is denoted by ʈuʈ and is n and Distance in C ʈuʈ ϭ ͑u и u͒1͞2.

The Euclidean distance between u and v is d͑u, v͒ ϭ ʈu Ϫ vʈ .

The Euclidean norm and distance may be expressed in terms of components as ʈ ʈ ϭ ͉͑ ͉ 2 ϩ ͉ ͉ 2 ϩ и и и ϩ ͉ ͉ 2͒1͞2 u u1 u2 un ͑ ͒ ϭ ͉͑ Ϫ ͉ 2 ϩ ͉ Ϫ ͉ 2 ϩ и и и ϩ ͉ Ϫ ͉ 2͒1͞2 d u, v u1 v1 u2 v2 un vn .

EXAMPLE 6 Finding the Euclidean Norm and Distance in C n Determine the norms of the vectors u ϭ ͑2 ϩ i, 0, 4 Ϫ 5i͒ and v ϭ ͑1 ϩ i, 2 ϩ i, 0͒ and find the distance between u and v.

Solution The norms of u and v are given as follows. ʈ ʈ ϭ ͉͑ ͉ 2 ϩ ͉ ͉ 2 ϩ ͉ ͉ 2͒1͞2 u u1 u2 u3 ϭ ͓͑22 ϩ 12͒ ϩ ͑02 ϩ 02͒ ϩ ͑42 ϩ 52͔͒1͞2 ϭ ͑5 ϩ 0 ϩ 41͒1͞2 ϭ ͙46

ʈ ʈ ϭ ͉͑ ͉ 2 ϩ ͉ ͉ 2 ϩ ͉ ͉ 2͒1͞2 v v1 v2 v3 ϭ ͓͑12 ϩ 12͒ ϩ ͑22 ϩ 12͒ ϩ ͑02 ϩ 02͔͒1͞2 ϭ ͑2 ϩ 5 ϩ 0͒1͞2 ϭ ͙7

The distance between u and v is given by d ͑u, v͒ ϭ ʈu Ϫ vʈ ϭ ʈ͑1, Ϫ2 Ϫ i, 4 Ϫ 5i͒ʈ ϭ ͓͑12 ϩ 02͒ ϩ ͑͑Ϫ2͒2 ϩ ͑Ϫ1͒2͒ ϩ ͑42 ϩ 52͔͒1͞2 ϭ ͑1 ϩ 5 ϩ 41͒1͞2 ϭ ͙47. 460 CHAPTER 8 COMPLEX VECTOR SPACES

Complex Inner Product Spaces The Euclidean inner product is the most commonly used inner product in Cn . However, on occasion it is useful to consider other inner products. To generalize the notion of an inner product, we use the properties listed in Theorem 8.7.

Definition of a Complex Let u and v be vectors in a complex vector space. A function that associates with u and Inner Product v the complex number Όu, v΍ is called a complex inner product if it satisfies the fol- lowing properties. 1. Όu, v΍ ϭ Όv, u΍ 2. Όu ϩ v, w΍ ϭ Όu, w΍ ϩ Όv, w΍ 3. Όku, v΍ ϭ kΌu, v΍ 4. Όu, u΍ ≥ 0 and Όu, u΍ ϭ 0 if and only if u ϭ 0.

A complex vector space with a complex inner product is called a complex inner product space or unitary space.

EXAMPLE 7 A Complex Inner Product Space

ϭ ͑ ͒ ϭ ͑ ͒ 2 Let u u1, u2 andv v1, v2 be vectors in the complex spaceC . Show that the func- tion defined by ͗ ͘ ϭ ϩ u, v u1v1 2u2v2 is a complex inner product.

Solution We verify the four properties of a complex inner product as follows. Ό ΍ ϭ ϩ ϭ ϩ ϭ ͗ ͘ 1. v, u v1u1 2v2u2 u1v1 2u2v2 u, v ͗ ϩ ͘ ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ 2. u v, w u1 v1 w1 2 u2 v2 w2 ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ u1w1 2u2w2 v1w1 2v2w2 ϭ ͗u, w͘ ϩ ͗v, w͘ ͗ ͘ ϭ ͑ ͒ ϩ ͑ ͒ ϭ ͑ ϩ ͒ ϭ ͗ ͘ 3. ku, v ku1 v1 2 ku2 v2 k u1v1 2u2v2 k u, v ͗ ͘ ϭ ϩ ϭ ͉ ͉2 ϩ ͉ ͉2 Ն 4. u, u u1u1 2u 2u2 u1 2 u2 0 ͗ ͘ ϭ ϭ ϭ Moreover, u, u 0 if and only if u1 u2 0. Since all properties hold,͗u, v͘ is a complex inner product. SECTION 8.4 EXERCISES 461

SECTION 8.4 ❑ EXERCISES

In Exercises 1–8, perform the indicated operation using In Exercises 35–38, determine whether the given function is a com- ϭ ͑ ͒ ϭ ͑ ͒ u ϭ ͑i, 3 Ϫ i͒, v ϭ ͑2 ϩ i, 3 ϩ i͒, and w ϭ ͑4i, 6͒. plex inner product, where u u1, u2 and v v1, v2 . ͗ ͘ ϭ ϩ 1.3u 2. 4iw 35. u, v u1 u2v2 ͑ ϩ ͒ ϩ ͗ ͘ ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ 3.1 2i w 4. iv 3w 36. u, v u1 v1 2 u2 v2 5.u Ϫ ͑2 Ϫ i͒v 6. ͑6 ϩ 3i͒v Ϫ ͑2 ϩ 2i͒w ͗ ͘ ϭ ϩ ͗ ͘ ϭ Ϫ 37.u, v 4u1v1 6u2v2 38. u, v u1v1 u2v2 7.u ϩ iv ϩ 2iw 8. 2iv Ϫ ͑3 Ϫ i͒w ϩ u ϭ ͑ ͒ ϭ ͑ ͒ ϭ ͑ ͒ 39. Let v1 i, 0, 0 and v2 i, i, 0 . If v3 z1, z2, z3 and the 3 In Exercises 9–12, determine whether S is a basis for Cn. set {v1, v2, v3} is not a basis for C , what does this imply about z1, z2, and z3? 9.S ϭ ͕͑1, i͒, ͑i, Ϫ1͖͒ 10. S ϭ ͕͑1, i͒, ͑i, 1͖͒ ϭ ͑ ͒ ϭ ͑ ͒ 40. Let v1 i, i, i and v2 1, 0, 1 . Determine a vector v3 11. S ϭ ͕͑i, 0, 0͒, ͑0, i, i͒, ͑0, 0, 1͖͒ ͕ ͖ 3 such that v1, v2, v3 is a basis for C . 12. S ϭ ͕͑1 Ϫ i, 0, 1͒, ͑2, i, 1 ϩ i͒, ͑1 Ϫ i, 1, 1͖͒ In Exercises 41–45, prove the given property where u, v, and w are In Exercises 13–16, express v as a linear combination of the vectors in C n and k is a complex number. following basis vectors. 41.͑u ϩ v͒и w ϭ uи w ϩ v и w 42. ͑ku)и v ϭ k͑u и v͒ (a) ͕͑i, 0, 0͒, ͑i, i, 0͒, ͑i, i, i͖͒ 43.u и ͑kv͒ ϭ k͑u и v͒ 44. uи u Ն 0 (b) ͕͑1, 0, 0͒, ͑1, 1, 0͒, ͑0, 0, 1 ϩ i͖͒ 45. u и u ϭ 0 if and only if u ϭ 0. 13.v ϭ ͑1, 2, 0͒ 14. v ϭ ͑1 Ϫ i, 1 ϩ i, Ϫ3͒ 46. Let ͗u, v͘ be a complex inner product and k a complex ϭ ͑Ϫ ϩ Ϫ ͒ ϭ ͑ ͒ 15.v i, 2 i, 1 16. v i, i, i number. How are ͗u, v͘ and ͗u, kv͘ related? In Exercises 17–24, determine the Euclidean norm of v. In Exercises 47 and 48, determine the linear transformation T : Cm → Cn that has the given characteristics. 17.v ϭ ͑i, Ϫi͒ 18. v ϭ ͑1, 0͒ 19.v ϭ 3͑6 ϩ i, 2 Ϫ i͒ 20. v ϭ ͑2 ϩ 3i, 2 Ϫ 3i͒ 47. T͑1, 0͒ ϭ ͑2 ϩ i, 1͒, T͑0, 1͒ ϭ ͑0, Ϫi͒ 21.v ϭ ͑1, 2 ϩ i, Ϫi͒ 22. v ϭ ͑0, 0, 0͒ 48. T͑i, 0) ϭ ͑2 ϩ i, 1͒, T͑0, i͒ ϭ ͑0, Ϫi͒ 23. v ϭ ͑1 Ϫ 2i, i, 3i, 1 ϩ i͒ In Exercises 49–52, the linear transformation T : Cm → C n is given ϭ ͑ Ϫ ϩ Ϫ ͒ 24. v 2, 1 i, 2 i, 4i byT͑v͒ ϭ Av. Find the image of v and the preimage of w. In Exercises 25–30, determine the Euclidean distance between u 1 0 1 ϩ i 0 and v. 49. A ϭ ΄ ΅, v ϭ ΄ ΅, w ϭ ΄ ΅ i i 1 Ϫ i 0 ϭ ͑ ͒ ϭ ͑ ͒ 25. u 1, 0 , v i, i i ϭ ͑ ϩ Ϫ ͒ ϭ ͑ ϩ Ϫ ͒ 0 i 1 1 26. u 2 i, 4, i , v 2 i, 4, i 50. A ϭ ΄ ΅, v ϭ 0 , w ϭ ΄ ΅ ϭ ͑ ͒ ϭ ͑ ͒ i 0 0 ΄ ΅ 1 27. u i, 2i, 3i , v 0, 1, 0 1 ϩ i ϭ ͑ Ϫ ͒ ϭ ͑ ͒ 28. u ͙2, 2i, i , v i, i, i 1 0 2 2 Ϫ i 29. u ϭ ͑1, 0͒, v ϭ ͑0, 1͒ 51. A ϭ i 0 , v ϭ ΄ ΅, w ϭ 2i ΄ ΅ 3 ϩ 2i ΄ ΅ 30. u ϭ ͑1, 2, 1, Ϫ2i͒, v ϭ ͑i, 2i, i, 2͒ i i 3i In Exercises 31–34, determine whether the set of vectors is linearly 0 1 1 2 1 Ϫ i independent or linearly dependent. 52. A ϭ ΄ i i Ϫ1΅, v ϭ ΄5΅, w ϭ ΄1 ϩ i΅ 31. ͕͑1, i͒, ͑i, Ϫ1͖͒ 0 i 0 0 i 32. ͕͑1 ϩ i, 1 Ϫ i, 1͒, ͑i, 0, 1͒, ͑Ϫ2, Ϫ1 ϩ i, 0͖͒ 53. Find the kernel of the linear transformation given in Exercise 33. ͕͑1, i, 1 ϩ i͒, ͑0, i, Ϫi͒, ͑0, 0, 1͖͒ 49. 34. ͕͑1 ϩ i, 1 Ϫ i, 0͒, ͑1 Ϫ i, 0, 0͒, ͑0, 1, 1͖͒ 462 CHAPTER 8 COMPLEX VECTOR SPACES

54. Find the kernel of the linear transformation given in Exercise 50. 58. Determine which of the following sets are subspaces of the vector space of complex-valued functions (see Example 4). In Exercises 55 and 56, find the image of v ϭ ͑i, i͒ for the indicated (a) The set of all functions f satisfying f ͑i͒ ϭ 0. composition, where T1 and T2 are given by the following matrices. (b) The set of all functions f satisfying f ͑0͒ ϭ 1. 0 i Ϫi i ϭ and A ϭ ͑ ͒ ϭ ͑Ϫ ͒ A1 ΄ ΅ 2 ΄ ΅ (c) The set of all functions f satisfying f.i f i i 0 i Ϫi

55.T2 Њ T1 56. T1 Њ T2 57. Determine which of the following sets are subspaces of the vector space of 2 ϫ 2 complex matrices. (a) The set of 2 ϫ 2 symmetric matrices. (b) The set of 2 ϫ 2 matrices A satisfying ͑A͒T ϭ A. (c) The set of 2 ϫ 2 matrices in which all entries are real. (d) The set of 2 ϫ 2 diagonal matrices.

8.5 UNITARY AND HERMITIAN MATRICES

Problems involving diagonalization of complex matrices, and the associated eigenvalue problems, require the concept of unitary and Hermitian matrices. These matrices roughly correspond to orthogonal and symmetric real matrices. In order to define unitary and Hermitian matrices, we first introduce the concept of the conjugate transpose of a com- plex matrix.

Definition of the The conjugate transpose of a complex matrix A, denoted by A*, is given by Conjugate Transpose of a A* ϭ A T Complex Matrix where the entries ofA are the complex conjugates of the corresponding entries of A.

Note that if A is a matrix with real entries, then A*ϭ AT . To find the conjugate trans- pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example.

EXAMPLE 1 Finding the Conjugate Transpose of a Complex Matrix Determine A* for the matrix 3 ϩ 7i 0 A ϭ ΄ ΅. 2i 4 Ϫ i