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 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 463
Solution 3 ϩ 7i 0 3 Ϫ 7i 0 A ϭ ϭ [ 2 i 4 Ϫ i ] [ Ϫ 2 i 4 ϩ i ] 3 Ϫ 7i Ϫ2i A* ϭ AT ϭ [ 0 4 ϩ i ]
We list several properties of the conjugate transpose of a matrix in the following theo- rem. The proofs of these properties are straightforward and are left for you to supply in Exercises 49–52.
Theorem 8.8 If A and B are complex matrices and k is a complex number, then the following proper- ties are true. Properties of 1. (A*)* = A Conjugate Transpose 2. (A+ B)* = A* + B* 3. (kA)* = kA * 4. (AB)* = B*A*
Unitary Matrices Recall that a real matrix A is orthogonal if and only if AϪ1 ϭ AT. In the complex system, matrices having the property that AϪ1 ϭ A * are more useful and we call such matrices unitary.
Definition of a A complex matrix A is called unitary if Unitary Matrix AϪ1 ϭ A*.
EXAMPLE 2 A Unitary Matrix Show that the following matrix is unitary. 1 1 ϩ i 1 Ϫ i A ϭ ΄ ΅ 2 1 Ϫ i 1 ϩ i
Solution Since ϩ Ϫ Ϫ ϩ ϭ 1 1 i 1 i 1 1 i 1 i ϭ 1 4 0 ϭ AA* ΄ ΅ ΄ ΅ ΄ ΅ I2 2 1 Ϫ i 1 ϩ i 2 1 ϩ i 1 Ϫ i 4 0 4
we conclude that A*ϭ AϪ1. Therefore, A is a unitary matrix. 464 CHAPTER 8 COMPLEX VECTOR SPACES
In Section 7.3, we showed that a real matrix is orthogonal if and only if its row (or column) vectors form an orthonormal set. For complex matrices, this property characterizes matrices that are unitary. Note that we call a set of vectors ͕ ͖ v1, v2, . . . , vm in Cn (complex Euclidean space) orthonormal if the following are true. ʈ ʈ ϭ ϭ 1. vi 1, i 1, 2, . . . , m и ϭ Þ 2. vi vj 0, i j The proof of the following theorem is similar to the proof of Theorem 7.8 given in Section 7.3.
Theorem 8.9 Ann ϫ n complex matrix A is unitary if and only if its row (or column) vectors form n Unitary Matrices an orthonormal set in C .
EXAMPLE 3 The Row Vectors of a Unitary Matrix Show that the following complex matrix is unitary by showing that its set of row vectors form an orthonormal set in C 3. ϩ 1 1 i Ϫ1 2 2 2 i i 1 A ϭ Ϫ ͙3 ͙3 ͙3 5i 3 ϩ i 4 ϩ 3i 2͙15 2͙15 2͙15
Solution We letr1, r2, andr3 be defined as follows. ϩ ϭ 1 1 i Ϫ1 r1 , , 2 2 2
ϭ Ϫ i i 1 r2 , , ͙3 ͙3 ͙3 ϩ ϩ ϭ 5i 3 i 4 3i r3 , , 2͙15 2͙15 2͙15
The length of r1 is ʈ ʈ ϭ ͑ и ͒1͞2 r1 r1 r1 ͞ 1 1 1 ϩ i 1 ϩ i Ϫ1 Ϫ1 1 2 ϭ ΄ ϩ ϩ ΅ 2 2 2 2 2 2 1 2 1 1͞2 ϭ ΄ ϩ ϩ ΅ ϭ 1. 4 4 4 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 465
The vectors r2 and r3 can also be shown to be unit vectors. The inner product of r1 and r2 is given by Ϫ ϩ Ϫ и ϭ 1 i ϩ 1 i i ϩ 1 1 r1 r2 2 ͙3 2 ͙3 2 ͙3 1 i 1 ϩ i Ϫi Ϫ1 1 ϭ ϩ ϩ 2 ͙3 2 ͙3 2 ͙3 i i 1 1 ϭ Ϫ ϩ Ϫ 2͙3 2͙3 2͙3 2͙3 ϭ 0.
и ϭ и ϭ ͕ ͖ Similarly, r1 r3 0 and r2 r3 0 and we can conclude that r1, r2, r3 is an orthonormal set. (Try showing that the column vectors of A also form an orthonormal set in C3.)
Hermitian Matrices A real matrix is called symmetric if it is equal to its own transpose. In the complex system, the more useful type of matrix is one that is equal to its own conjugate transpose. We call such a matrix Hermitian after the French mathematician Charles Hermite (1822–1901).
Definition of a A square matrix A is Hermitian if Hermitian Matrix A ϭ A*.
As with symmetric matrices, we can easily recognize Hermitian matrices by inspection. To see this, consider the 2 ϫ 2 matrix. a ϩ a i b ϩ b i ϭ 1 2 1 2 A ΄ ϩ ϩ ΅. c1 c2i d1 d2i The conjugate transpose of A has the form A* ϭ AT a ϩ a i c ϩ c i ϭ 1 2 1 2 A* ΄ ϩ ϩ ΅ b1 b2i d1 d2i a Ϫ a i c Ϫ c i ϭ 1 2 1 2 A* ΄ Ϫ Ϫ ΅. b1 b2i d1 d2i
If A is Hermitian, then A ϭ A * and we can conclude that A must be of the form a b ϩ b i ϭ 1 1 2 A ΄ Ϫ ΅. b1 b2i d1 466 CHAPTER 8 COMPLEX VECTOR SPACES
Similar results can be obtained for Hermitian matrices of order n ϫ n. In other words, a square matrix A is Hermitian if and only if the following two conditions are met. 1. The entries on the main diagonal of A are real.
2. The entryaij in the ith row and the jth column is the complex conjugate of the entry aji in the jth row and ith column.
EXAMPLE 4 Hermitian Matrices Which of the following matrices are Hermitian?
1 3 Ϫ i 0 3 Ϫ 2i (a)΄ ΅ (b) ΄ ΅ 3 ϩ i i 3 Ϫ 2i 4
3 2 Ϫ i Ϫ3i Ϫ1 2 3 (c)΄2 ϩ i 0 1 Ϫ i ΅ (d) ΄ 2 0 Ϫ1΅ 3i 1 ϩ i 0 3 Ϫ1 4
Solution (a) This matrix is not Hermitian because it has an imaginary entry on its main diagonal. (b) This matrix is symmetric but not Hermitian because the entry in the first row and second column is not the complex conjugate of the entry in the second row and first column. (c) This matrix is Hermitian. (d) This matrix is Hermitian, because all real symmetric matrices are Hermitian.
One of the most important characteristics of Hermitian matrices is that their eigenvalues are real. This is formally stated in the next theorem.
Theorem 8.10 If A is a Hermitian matrix, then its eigenvalues are real numbers. The Eigenvalues of a Hermitian Matrix
Proof Let be an eigenvalue of A and ϩ a1 b1i a ϩ b i v ϭ 2 2 ΄ . ΅ . ϩ an bni be its corresponding eigenvector. If we multiply both sides of the equation Av ϭ v by the row vector v*, we obtain ϭ ͑ ͒ ϭ ͑ ͒ ϭ ͑ 2 ϩ 2 ϩ 2 ϩ 2 ϩ и и и ϩ 2 ϩ 2͒ v**Av v v v *v a1 b1 a2 b2 an bn . Furthermore, since ͑v**Av͒ ϭ v *A*(v*)* ϭ v*Av SECTION 8.5 UNITARY AND HERMITIAN MATRICES 467
it follows that v*Av is a Hermitian 1 ϫ 1 matrix. This implies that v*Av is a real number, and we may conclude that is real.
REMARK: Note that this theorem implies that the eigenvalues of a real symmetric matrix are real, as stated in Theorem 7.7.
To find the eigenvalues of complex matrices, we follow the same procedure as for real matrices.
EXAMPLE 5 Finding the Eigenvalues of a Hermitian Matrix Find the eigenvalues of the following matrix. 3 2 Ϫ i Ϫ3i A ϭ ΄2 ϩ i 0 1 Ϫ i ΅ 3i 1 ϩ i 0
Solution The characteristic polynomial of A is
Ϫ 3 Ϫ2 ϩ i 3i ͉I Ϫ A͉ ϭ ΄Ϫ2 Ϫ i Ϫ1 ϩ i ΅ Ϫ3i Ϫ1 Ϫ i
ϭ ͑ Ϫ 3͒͑2 Ϫ 2͒ Ϫ ͑Ϫ2 ϩ i͓͒͑Ϫ2 Ϫ i͒ Ϫ ͑3i ϩ 3͔͒ ϩ 3i ͓͑1 ϩ 3i͒ ϩ 3i͔ ϭ ͑3 Ϫ 32 Ϫ 2 ϩ 6͒ Ϫ ͑5 ϩ 9 ϩ 3i͒ ϩ ͑3i Ϫ 9 Ϫ 9͒ ϭ 3 Ϫ 32 Ϫ 16 Ϫ 12 ϭ ͑ ϩ 1͒͑ Ϫ 6͒͑ ϩ 2͒ which implies that the eigenvalues of A are Ϫ1, 6, and Ϫ2.
To find the eigenvectors of a complex matrix, we use a similar procedure to that used for a real matrix. For instance, in Example 5, the eigenvector corresponding to the eigenvalue ϭϪ1 is obtained by solving the following equation. Ϫ Ϫ ϩ 3 2 i 3i v1 0 Ϫ2 Ϫ i Ϫ 1 ϩ i v ϭ 0 ΄ ΅΄ 2΅ ΄ ΅ Ϫ Ϫ Ϫ 3i 1 i v3 0 Ϫ Ϫ ϩ 4 2 i 3i v1 0 Ϫ2 Ϫ i Ϫ1 Ϫ 1 ϩ i v ϭ 0 ΄ ΅΄2΅ ΄ ΅ Ϫ Ϫ Ϫ Ϫ 3i 1 i 1 v3 0 468 CHAPTER 8 COMPLEX VECTOR SPACES
Using Gauss-Jordan elimination, or a computer or calculator, we obtain the following ϭϪ eigenvector corresponding to 1 1. Ϫ1 ͑ ϭϪ ͒ v ϭ 1 ϩ 2i 1 1 1 ΄ ΅ 1 ϭ ϭϪ Eigenvectors for 2 6 and 3 2 can be found in a similar manner. They are 1 Ϫ 21i 1 ϩ 3i 6 Ϫ 9i ͑ ϭ 6͒; Ϫ2 Ϫ i ͑ ϭϪ2͒ ΄ ΅ 2 ΄ ΅ 3 13 5
TECHNOLOGY Some computers and calculators have built-in programs for finding the eigenvalues and NOTE corresponding eigenvectors of complex matrices. For example, on the TI-85, the eigVl key on the MATRX MATH menu calculates the eigenvalues of the matrix A, and the eigVc key gives the corresponding eigenvectors.
Just as we saw in Section 7.3 that real symmetric matrices were orthogonally diagonal- izable, we will show now that Hermitian matrices are unitarily diagonalizable. A square matrix A is unitarily diagonalizable if there exists a unitary matrix P such that PϪ1AP is a diagonal matrix. Since P is unitary, PϪ1 ϭ P *, so an equivalent statement is that A is unitarily diagonalizable if there exists a unitary matrix P such that P *AP is a diagonal matrix. The next theorem tells us that Hermitian matrices are unitarily diagonalizable.
Theorem 8.11 If A is an n ϫ n Hermitian matrix, then 1. eigenvectors corresponding to distinct eigenvalues are orthogonal. Hermitian Matrices and 2. A is unitarily diagonalizable. Diagonalization
Proof To prove part 1, letv1 and v2 be two eigenvectors corresponding to the distinct (and real) ϭ ϭ eigenvalues 1 and 2 . Because Av1 1v1 and Av2 2v2, we have the following equa- ͑ ͒ tions for the matrix product Av1 *v2. ͑ ͒ ϭ ϭ ϭ ϭ Av1 **v2 v1 A**v2 v1 Av2 v1 *2v2 2v1 *v2 ͑ ͒ ϭ ͑ ͒ ϭ ϭ Av1 ****v2 1v1 v2 v1 1v2 1v1 v2 Therefore, Ϫ ϭ 2v1**v2 1v1 v2 0 ͑ Ϫ ͒ ϭ 2 1 v1*v2 0 ϭ Þ v1*v2 0 since 1 2,
and we have shown that v1 and v2 are orthogonal. Part 2 of Theorem 8.11 is often called the Spectral Theorem, and its proof is omitted. SECTION 8.5 UNITARY AND HERMITIAN MATRICES 469
EXAMPLE 6 The Eigenvectors of a Hermitian Matrix The eigenvectors of the Hermitian matrix given in Example 5 are mutually orthogonal because the eigenvalues are distinct. We can verify this by calculating the Euclidean inner и и и products v1 v2, v1 v3 and v2 v3. For example, и ϭ ͑Ϫ ͒͑ Ϫ ͒ ϩ ͑ ϩ ͒͑ Ϫ ͒ ϩ ͑ ͒͑ ͒ v 1 v2 1 1 21i 1 2i 6 9i 1 13 ϭ ͑Ϫ1͒͑1 ϩ 21i͒ ϩ ͑1 ϩ 2i͒͑6 ϩ 9i͒ ϩ 13 ϭϪ1 Ϫ 21i ϩ 6 ϩ 12i ϩ 9i Ϫ 18 ϩ 13 ϭ 0. и и The other two inner productsv1 v3 and v2 v3 can be shown to equal zero in a similar manner.
The three eigenvectors in Example 6 are mutually orthogonal because they correspond to distinct eigenvalues of the Hermitian matrix A. Two or more eigenvectors corresponding to the same eigenvector may not be orthogonal. However, once we obtain any set of linearly independent eigenvectors for a given eigenvalue, we can use the Gram-Schmidt orthonor- malization process to obtain an orthogonal set.
EXAMPLE 7 Diagonalization of a Hermitian Matrix Find a unitary matrix P such that P*AP is a diagonal matrix where 3 2 Ϫ i Ϫ3i A ϭ ΄2 ϩ i 0 1 Ϫ i ΅ . 3i 1 ϩ i 0 Solution The eigenvectors of A are given after Example 5. We form the matrix P by normalizing these three eigenvectors and using the results to create the columns of P. Thus, since ʈ ʈ ϭ ʈ͑Ϫ ϩ ͒ʈ ϭ ͙ ϩ ϩ ϭ ͙ v1 1, 1 2i, 1 1 5 1 7 ʈ ʈ ϭ ʈ͑ Ϫ Ϫ ͒ʈ ϭ ͙ ϩ ϩ ϭ ͙ v2 1 21i, 6 9i, 13 442 117 169 728 ʈ ʈ ϭ ʈ͑ ϩ Ϫ Ϫ ͒ʈ ϭ ͙ ϩ ϩ ϭ ͙ v3 1 3i, 2 i, 5 10 5 25 40 we obtain the unitary matrix P, 1 1 Ϫ 21i 1 ϩ 3i Ϫ ͙7 ͙728 ͙40 1 ϩ 2i 6 Ϫ 9i Ϫ2 Ϫ i . P = ͙7 ͙728 ͙40 1 13 5
͙7 ͙728 ͙40 470 CHAPTER 8 COMPLEX VECTOR SPACES
Try computing the product P*AP for the matrices A and P in Example 7 to see that you obtain Ϫ1 0 0 P*AP ϭ ΄ 0 6 0΅ 0 0 Ϫ2 where Ϫ1, 6, and Ϫ2 are the eigenvalues of A. We have seen that Hermitian matrices are unitarily diagonalizable. However, it turns out that there is a larger class of matrices, called normal matrices, which are also unitarily di- agonalizable. A square complex matrix A is normal if it commutes with its conjugate trans- pose: AA* = A*A. The main theorem of normal matrices says that a complex matrix A is normal if and only if it is unitarily diagonalizable. You are asked to explore normal matri- ces further in Exercise 59. The properties of complex matrices described in this section are comparable to the properties of real matrices discussed in Chapter 7. The following summary indicates the correspondence between unitary and Hermitian complex matrices when compared with or- thogonal and symmetric real matrices.
Comparison of A is a symmetric matrix A is a Hermitian matrix (Real) (Complex) Hermitian and 1. Eigenvalues of A are real. 1. Eigenvalues of A are real. Symmetric Matrices 2. Eigenvectors corresponding to 2. Eigenvectors corresponding to distinct eigenvalues are distinct eigenvalues are orthogonal. orthogonal. 3. There exists an orthogonal 3. There exists a unitary matrix matrix P such that P such that PTAP P*AP is diagonal. is diagonal. SECTION 8.5 EXERCISES 471
SECTION 8.5 ❑ EXERCISES
In Exercises 1– 8, determine the conjugate transpose of the given i i i Ϫ matrix. ͙2 ͙3 ͙6 4 3 Ϫ i Ϫi 1 ϩ 2i 2 Ϫ i i i i 5 5 1.A ϭ ΄ ΅ 2. A ϭ ΄ ΅ 17.A ϭ 18. A ϭ 2 3i 1 1 ͙ ͙ ͙ 3 4 2 3 6 i i 5 5 0 1 4 ϩ 3i 2 ϩ i i Ϫ i 3.A ϭ ΄ ΅ 4. A ϭ ΄ ΅ 0 2 0 2 Ϫ i 6i ͙3 ͙6
0 5 ϩ i ͙2i In Exercises 19–22. (a) verify that A is unitary by showing that its 5. A ϭ ΄ 5 Ϫ i 6 4 ΅ rows are orthonormal, and (b) determine the inverse of A. Ϫ͙2i 4 3 4 3 1 ϩ i 1 ϩ i Ϫ i Ϫ 5 5 2 2 7 ϩ 5i 19.A ϭ 20. A ϭ 2 ϩ i 3 Ϫ i 4 ϩ 5i 3 4 1 1 6.A ϭ ΄ ΅ 7. A ϭ 2i i 3 Ϫ i 2 6 Ϫ 2i ΄ ΅ 5 5 ͙2 ͙2 4 2 i 1 ͙3 Ϫ i 1 ϩ ͙3i 21. A ϭ 8. A ϭ ΄5 3i ΅ 2͙2 [͙ 3 ϩ i 1 Ϫ ͙ 3 i ] 0 6 Ϫ i 0 1 0 In Exercises 9–12, explain why the given matrix is not unitary. Ϫ1 ϩ i 1 Ϫ i 0 i 0 1 i 22. A ϭ ͙6 ͙3 9.A ϭ ΄ ΅ 10. A ϭ ΄ ΅ 0 0 i Ϫ1 2 1 0 ͙ ͙ 1 ϩ i i 6 3 0 Ϫ ͙2 ͙2 In Exercises 23–28, determine whether the matrix A is Hermitian. 11. A ϭ 0 1 0 0 2 ϩ i 1 ϭ Ϫ ϭ 1 0 1 1 1 ϩ i 23.A ΄2 i i 0΅ 24. A ΄ ΅ Ϫ 0 1 2 2 2 1 0 1 i 1 i 12. A ϭ Ϫ 0 i ͙3 ͙3 ͙3 25. A ϭ ΄ ΅ Ϫi 0 1 1 1 ϩ i Ϫ Ϫ 2 2 2 ϩ Ϫ ϭ 1 2 i 3 i 26. A ΄ Ϫ ϩ ΅ In Exercises 13–18, determine whether A is unitary by calculating 2 i 2 3 i AA*. ϭ 0 0 1 ϩ i 1 ϩ i 1 ϩ i 1 Ϫ i 27. A ΄ ΅ 13.A ϭ ΄ ΅ 14. A ϭ ΄ ΅ 0 0 1 Ϫ i 1 Ϫ i 1 Ϫ i 1 ϩ i 1 ͙2 ϩ i 5 i i 28. A ϭ ͙2 Ϫ i 2 3 ϩ i ͙ ͙ ΄ ΅ 15.A ϭ I 16. A ϭ 2 2 5 3 Ϫ i 6 n i i Ϫ ͙2 ͙2 472 CHAPTER 8 COMPLEX VECTOR SPACES
In Exercises 29–34, determine the eigenvalues of the matrix A. 44. Let z be a complex number with modulus 1. Show that the following matrix is unitary. 0 i 0 2 ϩ i 29.A ϭ ΄ ΅ 30. A ϭ ΄ ΅ 1 z z Ϫi 0 2 Ϫ i 4 A ϭ ͙2 [ iz Ϫ iz ] Ϫ ϭ 3 1 i ϭ 3 i In Exercises 45–48, use the result of Exercise 44 to determine a, b, 31.A ΄ ϩ ΅ 32. A ΄Ϫ ΅ 1 i 2 i 3 and c so that A is unitary. 33Ϫ 4i i i 1 Ϫ1 a 1 a 2 Ϫ 45.A ϭ ΄ ΅ 46. A ϭ 5 ͙2 ͙2 ͙2 b c ͙2 ΄ ΅ b c i 33. A ϭ 2 0 ͙2 3 6 ϩ 3i 1 i a 1 a i 47.A ϭ ΄ ΅ 48. A ϭ ͙45 Ϫ 0 2 ͙2 b c ͙2 ΄ ΅ ͙2 bc
1 4 1 Ϫ i In Exercises 49–52, prove the given formula, where A and B are ϫ 34. A ϭ ΄0 i 3i ΅ n n complex matrices. ءϩ B ءϭ A ءϭ A 50. ͑A ϩ B͒ ء͒ءϩ i 49.͑A 2 0 0 ء ء ϭ ء͒ ͑ ء ϭ ء͒ ͑ In Exercises 35–38, determine the eigenvectors of the given matrix. 51.kA kA 52. AB B A ϩ A ϭ O. Prove that iA is ءLet A be a matrix such that A .53 35. The matrix in Exercise 29. Hermitian. 36. The matrix in Exercise 30. 54. Show that det͑A͒ ϭ det͑A͒, where A is a 2 ϫ 2 matrix. 37. The matrix in Exercise 33. In Exercises 55–56, assume that the result of Exercise 54 is true for 38. The matrix in Exercise 32. matrices of any size. In Exercises 39–43, find a unitary matrix P that diagonalizes the .ϭ det͑A͒ ͒ءShow that det͑A .55 given matrix A. 56. Prove that if A is unitary, then ͉det͑A͉͒ ϭ 1. 0 i 0 2 ϩ i 39.A ϭ ΄ ΅ 40. A ϭ ΄ ΅ 57. (a) Prove that every Hermitian matrix A can be written as the Ϫi 0 2 Ϫ i 4 sum A ϭ B ϩ iC, where B is a real symmetric matrix and C is real and skew-symmetric. i i (b) Use part (a) to write the matrix 2 Ϫ ͙2 ͙2 ϩ ϭ 2 1 i i A ΄ Ϫ ΅ 41. A ϭ 2 0 1 i 3 ͙2 as a sum A ϭ B ϩ iC, where B is a real symmetric matrix i Ϫ 0 2 and C is real and skew-symmetric. ͙2 (c) Prove that every n ϫ n complex matrix A can be written 4 2 ϩ 2i as A ϭ B ϩ iC, where B and C are Hermitian. 42. A ϭ ΄ ΅ 2 Ϫ 2i 6 (d) Use part (c) to write the complex matrix Ϫ1 0 0 i 2 A ϭ ΄ ΅ 43. A ϭ ΄ 0 Ϫ1 Ϫ1 ϩ i΅ 2 ϩ i 1 Ϫ 2i 0 Ϫ1 Ϫ i 0 as a sum A ϭ B ϩ iC, where B and C are Hermitian. CHAPTER 8 REVIEW EXERCISES 473
58. Determine which of the following sets are subspaces of the (d) Find a 2 ϫ 2 matrix that is unitary, but not Hermitian. ϫ vector space of n n complex matrices. (e) Find a 2 ϫ 2 matrix that is normal, but neither Hermitian (a) The set of n ϫ n Hermitian matrices. nor unitary. (b) The set of n ϫ n unitary matrices. (f) Find the eigenvalues and corresponding eigenvectors of (c) The set of n ϫ n normal matrices. your matrix from part (e). (g) Show that the complex matrix 59. (a) Prove that every Hermitian matrix is normal. ΄i 1 ΅ (b) Prove that every unitary matrix is normal. 0 i (c) Find a 2 ϫ 2 matrix that is Hermitian, but not unitary. is not diagonalizable. Is this matrix normal?
CHAPTER 8 ❑ REVIEW EXERCISES
In Exercises 1–6, perform the given operation. (1 Ϫ 2i͒(1 ϩ 2i͒ 5 ϩ 2i 23. 24. 1. Find u ϩ z : u ϭ 2 Ϫ 4i, z ϭ 4i 3 Ϫ 3i (Ϫ2 ϩ 2i)(2 Ϫ 3i) Ϫ ϭ ϭ 2. Find u z : u 4, z 8i In Exercises 25 and 26, find AϪ1 (if it exists). 3. Find uz : u ϭ 4 Ϫ 2i, z ϭ 4 ϩ 2i Ϫ Ϫ Ϫ 4. Find uz : u ϭ 2i, z ϭ 1 Ϫ 2i 3 i 1 2i 25. A ϭ 23 11 u ΄Ϫ ϩ ϩ ΅ 5. Find : u ϭ 6 Ϫ 2i, z ϭ 3 Ϫ 3i i 2 3i z 5 5 u 6. Find : u ϭ 7 ϩ i, z ϭ i 5 1 Ϫ i z 26. A ϭ ΄ ΅ 0 i In Exercises 7–10, find all zeros of the given polynomial. In Exercises 27–30, determine the polar form of the complex 7.x2 Ϫ 4x ϩ 8 8. x2 Ϫ 4x ϩ 7 number. 2 ϩ ϩ 3 ϩ 2 ϩ ϩ 9.3x 3x 3 10. x 2x 2x 1 27.4 ϩ 4i 28. 3 ϩ 2i In Exercises 11–14, perform the given operation using 29.7 Ϫ 4i 30. ͙3 ϩ i 4 Ϫ i 2 1 ϩ i i A ϭ ΄ ΅ and B ϭ ΄ ΅. In Exercises 31–34, find the standard form of the given complex 3 3 ϩ i 2i 2 ϩ i number. 11.A ϩ B 12. 2iB 31. 5΄cosϪ ϩ i sin Ϫ ΅ 13.det͑A Ϫ B͒ 14. 3BA 6 6 In Exercises 15–20, perform the given operation using w ϭ 5 5 32. 4cos ϩ i sin 2 Ϫ 2i, v ϭ 3 ϩ i, and z ϭϪ1 ϩ 2i. 4 4
15.z 16. v 2 2 3 3 33.6cos ϩ i sin 34. 7cos ϩ i sin 17.͉w͉ 18. ͉vz͉ 3 3 2 2 19.wv 20. ͉zw͉ In Exercises 35–38, perform the indicated operation. Leave the In Exercises 21–24, perform the indicated operation. result in polar form. 2 ϩ i 1 ϩ i 21. 22. 35. ΄4cos ϩ i sin ΅΄3cos ϩ i sin ΅ 2 Ϫ i Ϫ1 ϩ 2i 2 2 6 6 474 CHAPTER 8 COMPLEX VECTOR SPACES
1 56. v ϭ ͑2 ϩ i, Ϫ1 ϩ 2i, 3i͒, u ϭ ͑4 Ϫ 2i, 3 ϩ 2i, 4͒ 36. ΄ cos ϩ i sin ΅΄2cos Ϫ ϩ i sin Ϫ ΅ 2 2 2 2 2 In Exercises 57–60, determine whether the given matrix is unitary. ͞ ϩ ͞ 9[cos( 2) i sin ( 2)] i 1 2 ϩ i 1 ϩ i 37. ͞ ϩ ͞ Ϫ 6[cos(2 3) i sin (2 3)] ͙2 ͙2 4 4 57. 58. i 1 1 ͙2 4[cos(͞4) ϩ i sin (͞4)] 38. ͙2 ͙2 ͙ ͙ 7[cos(͞3) ϩ i sin (͞3)] 3 3 In Exercises 39– 42, find the indicated power of the given number 1 1 and express the result in polar form. 0 ͙2 ͙2 39.͑Ϫ1 Ϫ i͒4 40. ͑2i͒3 1 0 59.΄ ΅ 60. 0 i 0 i Ϫi 7 4 ͙ ϩ ϩ 1 ϩ i Ϫ1 Ϫ i 41.΄ 2cos i sin ΅ 42. ΄5cos i sin ΅ 0 6 6 3 3 2 2 In Exercises 43–46, express the given roots in standard form. In Exercises 61 and 62, determine whether the given matrix is 2 2 43. Square roots: 25cos ϩ i sin Hermitian. 3 3 Ϫ ϩ Ϫ 1 1 i 2 i 44. Cube roots: 27cos ϩ i sin 61. 1 Ϫ i 3 i 6 6 ΄ ΅ 2 ϩ i Ϫi 4 45. Cube roots: i 9 2 Ϫ i 2 46. Fourth roots: 16cos ϩ i sin 62. ΄2 ϩ i 0 Ϫ1 Ϫ i΅ 4 4 2 Ϫ1 ϩ i 3 In Exercises 47 and 48, determine the conjugate transpose of the given matrix. In Exercises 63 and 64, find the eigenvalues and corresponding eigenvectors of the given matrix. Ϫ1 ϩ 4i 3 ϩ i 47. A ϭ ΄ ΅ 2 0 Ϫi 3 Ϫ i 2 ϩ i 4 2 Ϫ i 63.΄ ΅ 64. 0 3 0 2 ϩ i 0 ΄ ΅ 5 2 Ϫ i 3 ϩ 2i i 0 2 ϭ ϩ Ϫ 48. A ΄2 2i 3 2i i ΅ 65. Prove that if A is an invertible matrix, then A* is also ϩ Ϫ Ϫ 3i 2 i 1 2i invertible. 66. Determine all complex numbers z such that z ϭϪz In Exercises 49–52, find the indicated vector using u ϭ ͑4i, 2 ϩ i͒, 67. Prove that if the product of two complex numbers is zero, v ϭ ͑3, Ϫi ͒, and w ϭ ͑3 Ϫ i, 4 ϩ i͒. then one of the numbers must be zero. 49.7u Ϫ v 50. 3iw ϩ ͑4 Ϫ i͒v 68. (a) Find the determinant of the following Hermitian matrix. ϩ Ϫ ͑ ϩ ͒ Ϫ ͑Ϫ ͒ 51.iu iv iw 52. 3 2i u 2i w 3 2 Ϫ i Ϫ3i In Exercises 53 and 54, determine the Euclidean norm of the given ΄2 ϩ i 0 1 Ϫ i ΅ vector. 3i 1 ϩ i 0
53.v ϭ ͑3 Ϫ 5i, 2i͒ 54. v ϭ ͑3i, Ϫ1 Ϫ 5i, 3 ϩ 2i͒ (b) Prove that the determinant of any Hermitian matrix is real. 69. Let A and B be Hermitian matrices. Prove that AB ϭ BA if In Exercises 55 and 56, find the Euclidean distance between the and only if AB is Hermitian. given vectors. Prove .ءLet u be a unit vector in Cn. Define H ϭ I Ϫ 2uu .70 55. v ϭ ͑2 Ϫ i, i͒, u ϭ ͑i, 2 Ϫ i͒ that H is an n ϫ n Hermitian and unitary matrix. CHAPTER 8 PROJECTS 475
71. Use mathematical induction to prove DeMoivre’s Theorem. 75. Prove that for all vectors u and v in a complex inner product 72. Prove that if z is a zero of a polynomial equation with real co- space, efficients, then the conjugate of z is also a zero. Όu, v ϭ 1 ʈu ϩ vʈ2 Ϫ ʈu Ϫ vʈ2 ϩ iʈu ϩ ivʈ2 ϩ 4[ 73. Show that if z1 z2 and z1z2 are both nonzero real numbers, Ϫ iʈu Ϫ ivʈ2 . then z1 and z2 are both real numbers. ] 74. Prove that if z and w are complex numbers, then ͉z ϩ w͉ Յ ͉z͉ ϩ ͉w͉.
CHAPTER 8 PROJECTS
1 Population Growth and Dynamical Systems - II In the projects for Chapter 7, you were asked to model the population of two species using a system of differential equations of the form Ј ͑ ͒ ϭ ͑ ͒ ϩ ͑ ͒ y 1 t ay1 t by2 t Ј ͑ ͒ ϭ ͑ ͒ ϩ ͑ ͒ y 2 t cy1 t dy2 t . The constants a, b, c, and d depend on the particular species being studied. In Chapter 7, we looked at an example of a predator–prey relationship, in which a ϭ 0.5, b ϭ 0.6, c ϭϪ0.4, and d ϭ 3.0. Suppose we now consider a slightly different model. Ј ͑ ͒ ϭ ͑ ͒ ϩ ͑ ͒ ͑ ͒ ϭ y 1 t 0.6y1 t 0.8y2 t , y1 0 36 Ј ͑ ͒ ϭϪ ͑ ͒ ϩ ͑ ͒ ͑ ͒ ϭ y 2 t 0.8y1 t 0.6y2 t , y2 0 121
͑ ͒ ͑ ͒ 1. Use the diagonalization technique to find the general solutions y1 t and y2 t at any time t > 0. Although the eigenvalues and eigenvectors of the matrix 0.6 0.8 A ϭ ΄ ΅ Ϫ0.8 0.6 are complex, the same principles apply, and you can obtain complex exponen- tial solutions. 2. Convert your complex solutions to real solutions by observing that if ϭ a ϩ bi is a (complex) eigenvalue of A with (complex) eigenvector v, then the real and imaginary parts of etv form a linearly independent pair of (real) solutions. You will need to use the formula ei ϭ cos ϩ i sin . 3. Use the initial conditions to find the explicit form of the (real) solutions to the original equations. 4. If you have access to a computer or graphing calculator, graph the solutions obtained in part (3) over the domain 0 ≤ t ≤ 3. At what moment are the two populations equal? 5. Interpret the solution in terms of the long-term population trend for the two species. Does one species ultimately disappear? Why or why not? Contrast this solution to that obtained for the model in Chapter 7. 6. If you have access to a computer or graphing calculator that can numerically solve differential equations, use it to graph the solutions to the original system of equations. Does this numerical approximation appear to be accurate?