8.5 Unitary and Hermitian Matrices

500 CHAPTER 8 COMPLEX VECTOR SPACES

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, the concept of the conjugate transpose of a complex matrix must first be introduced.

Definition of the The conjugate transpose of a complex matrix A, denoted by A*, is given by Conjugate Transpose of a A* 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*. AT To find the conjugate transpose of a matrix, 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 7i 0 A . 2i 4 i SECTION 8.5 UNITARY AND HERMITIAN MATRICES 501

Solution 3 7i 0 3 7i 0 A 2i 4 i 2i 4 i

3 7i 2i A* AT 0 4 i

Several properties of the conjugate transpose of a matrix are listed in the following theorem. The proofs of these properties are straightforward and are left for you to supply in Exercises 49Ð52.

Theorem 8.8 If A and B are complex matrices and k is a complex number, then the following properties 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 A1 AT. In the complex system, matrices having the property that A1 A * are more useful and such matrices are called unitary.

Definition of a A complex matrix A is unitary if Unitary Matrix A1 A*.

EXAMPLE 2 A Unitary Matrix Show that the matrix is unitary. 1 1 i 1 i A 2 1 i 1 1

Solution Because 1 1 i 1 i 1 1 i 1 i 1 4 0 AA* I , 2 1 i 1 i 2 1 i 1 i 4 0 4 2 you can conclude that A* A1. So, A is a unitary matrix. 502 CHAPTER 8 COMPLEX VECTOR SPACES

In Section 7.3, you saw 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 a set of vectors v1, v2, . . . , vm in Cn (complex Euclidean space) is called orthonormal if the following are true. 1. vi 1, i 1, 2, . . . , m 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 An n 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 complex matrix is unitary by showing that its set of row vectors form an orthonormal set in C 3. 1 1 i 1 2 2 2 i i 1 A 3 3 3 5i 3 i 4 3i 215 215 215

Solution Let r1, r2, and r3 be defined as follows. 1 1 i 1 r , , 1 2 2 2 i i 1 r , , 2 3 3 3 5i 3 i 4 3i r , , 3 215 215 215

The length of r1 is 12 r1 r1 r1 1 1 1 i 1 i 1 1 12 2 2 2 2 2 2 1 2 1 12 1. 4 4 4 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 503

The vectors r2 and r3 can also be shown to be unit vectors. The inner product of r1 and r2 is given by 1 i 1 i i 1 1 r r 1 2 2 3 2 3 2 3 1 i 1 i i 1 1 2 3 2 3 2 3 i i 1 1 23 23 23 23 0. Similarly,r1 r3 0 and r2 r3 0. So, you can conclude that r1, r2, r3 is an orthonormal set. Try showing that the column vectors of A also form an orthonormal set in C 3.

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. Such a matrix is called Hermitian after the French mathematician Charles Hermite (1822Ð1901).

Definition of a A square matrix A is Hermitian if Hermitian Matrix A A*.

As with symmetric matrices, you can easily recognize Hermitian matrices by inspection. To see this, consider the 2 2 matrix A. a1 a2i b1 b2i A c1 c2i d1 d2i The conjugate transpose of A has the form A* AT a1 a2i c1 c2i b1 b2i d1 d2i a1 a2i c1 c2i . b1 b2i d1 d2i If A is Hermitian, then A A*. So, you can conclude that A must be of the form a1 b1 b2i A . b1 b2i d1 504 CHAPTER 8 COMPLEX VECTOR SPACES

Similar results can be obtained for Hermitian matrices of order n 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 entry aij 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 matrices are Hermitian? 1 3 i 0 3 2i (a) (b) 3 i i 3 2i 4 3 2 i 3i 1 2 3 (c)2 i 0 1 i (d) 2 0 1 3i 1 i 0 3 1 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 a1 b1i a b i v 2 2 . . an bni be its corresponding eigenvector. If both sides of the equation Av v are multiplied by the row vector v*, then 2 2 2 2 . . . 2 2 v*Av v* v v*v a1 b1 a2 b2 an bn . Furthermore, because v*Av* v*A*v** v*Av, SECTION 8.5 UNITARY AND HERMITIAN MATRICES 505

it follows that v*Av is a Hermitian 1 1 matrix. This implies that v*Av is a real number, thus 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, follow the same procedure as for real matrices.

EXAMPLE 5 Finding the Eigenvalues of a Hermitian Matrix Find the eigenvalues of the matrix A.

3 2 i 3i A 2 i 0 1 i 3i 1 i 0

Solution The characteristic polynomial of A is

3 2 i 3i I A 2 i 1 i 3i 1 i

32 2 2 i2 i 3i 3 3i 1 3i 3i 3 32 2 6 5 9 3i 3i 9 9 3 32 16 12 1 6 2. This implies that the eigenvalues of A are 1, 6, and 2.

To find the eigenvectors of a complex matrix, use a similar procedure to that used for a real matrix. For instance, in Example 5, the eigenvector corresponding to the eigenvalue 1 is obtained by solving the following equation. 3 2 i 3i v1 0 2 i 1 i v 0 2 3i 1 i v3 0 4 2 i 3i v1 0 2 i 1 1 i v 0 2 3i 1 i 1 v3 0 506 CHAPTER 8 COMPLEX VECTOR SPACES

Using Gauss-Jordan elimination, or a computer or calculator, obtain the following eigenvector corresponding to 1 1. 1 v 1 2i 1 1 1 1 Eigenvectors for 2 6 and 3 2 can be found in a similar manner. They are 1 21i 1 3i 6 9i and2 i, respectively. 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-86, the eigVl key on the matrix math menu calculates the eigenvalues of the matrix A, and the eigVc key gives the corresponding eigenvectors.

Just as you saw in Section 7.3 that real symmetric matrices were orthogonally diagonalizable, you will see now that Hermitian matrices are unitarily diagonalizable. A square matrix A is unitarily diagonalizable if there exists a unitary matrix P such that P1AP is a diagonal matrix. Because P is unitary,P1 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 states that Hermitian matrices are unitarily diagonalizable.

Theorem 8.11 If A is an n 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, let v1 and v2 be two eigenvectors corresponding to the distinct (and real) eigenvalues 1 and 2. Because Av1 1v1 and Av2 2v2, you have the following equa- tions for the matrix product Av1 *v2. Av1 **v2 v1 A**v2 v1 Av2 v1 *2v2 2v1 *v2 Av1 ****v2 1v1 v2 v1 1v2 1v1 v2 So, 2v1*v2 1v1*v2 0 2 1 v1*v2 0 v1*v2 0 because 1 2,

and this shows that v1 and v2 are orthogonal. Part 2 of Theorem 8.11 is often called the Spectral Theorem, and its proof is left to you. SECTION 8.5 UNITARY AND HERMITIAN MATRICES 507

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. You can verify this by calculating the Euclidean inner

products v1 v2, v1 v3, and v2 v3. For example, v1 v2 1 1 21i 1 2i 6 9i 1 13 11 21i 1 2i6 9i 13 1 21i 6 12i 9i 18 13 0.

The other two inner products v1 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 any set of linearly independent eigenvectors is obtained for a given eigenvalue, the Gram-Schmidt orthonormalization process can be used to find 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 i 3i A 2 i 0 1 i. 3i 1 i 0

Solution The eigenvectors of A are given after Example 5. Form the matrix P by normalizing these three eigenvectors and using the results to create the columns of P. So, because v1 1, 1 2i, 1 1 5 1 7 v2 1 21i, 6 9i, 13 442 117 169 728 v3 1 3i, 2 i, 5 10 5 25 40, the unitary matrix P is obtained. 1 1 21i 1 3i 7 728 40 1 2i 6 9i 2 i P 7 728 40 1 13 5 7 728 40 508 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 1 0 0 P*AP 0 6 0 0 0 2 where 1, 6, and 2 are the eigenvalues of A. You 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 diagonalizable. A square complex matrix A is normal if it commutes with its conjugate transpose:AA* A*A. The main theorem of normal matrices states that a complex matrix A is normal if and only if it is unitarily diagonalizable. You are asked to explore normal matrices 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 orthogonal 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 509

SECTION 8.5 ❑ EXERCISES

In Exercises 1Ð 8, determine the conjugate transpose of the matrix. i i i i i 1 2i 2 i 2 3 6 4 3 1.A 2. A 2 3i 1 1 i i i 5 5 17. A 18. A 3 4 2 3 6 i i 0 1 4 3i 2 i 5 5 3.A 4. A i i 2 0 2 i 6i 0 3 6 0 5 i 2i In Exercises 19Ð22. (a) verify that A is unitary by showing that its 5. A 5 i 6 4 rows are orthonormal, and (b) determine the inverse of A. 2i 4 3 4 3 1 i 1 i 2 i 3 i 4 5i i 6. A 5 5 2 2 3 i 2 6 2i 19.A 20. A 3 4 1 1 i 7 5i 2 i 5 5 2 2 7.A 2i 8. A 5 3i 1 3 i 1 3i 4 0 6 i 21. A 22 3 i 1 3i In Exercises 9Ð12, explain why the matrix is not unitary. i 0 1 i 010 9.A 10. A 0 0 i 1 1 i 1 i 0 1 i i 22. A 6 3 0 11. A 2 2 2 1 0 0 1 0 6 3 In Exercises 23Ð28, determine whether the matrix A is Hermitian. 1 1 1 i 2 2 2 0 2 i 1 1 0 i 1 i 23.A 2 i i 0 24. A 12. A 0 1 3 3 3 1 0 1 1 1 1 i 0 i 2 2 2 25. A i 0 In Exercises 13Ð18, determine whether A is unitary by calculating 1 2 i 3 i AA*. 26. A 2 i 2 3 i 1 i 1 i 1 i 1 i 13.A 14. A 0 0 1 i 1 i 1 i 1 i 27. A 0 0 i i 2 2 1 2 i 5 15.A I 16. A n i i 28. A 2 i 2 3 i 2 2 5 3 i 6 510 CHAPTER 8 COMPLEX VECTOR SPACES

In Exercises 29Ð34, determine the eigenvalues of the matrix A. 44. Let z be a complex number with modulus 1. Show that the matrix A is unitary. 0 i 0 2 i 29.A 30. A 1 z z i 0 2 i 4 A 2 iz iz 3 1 i 3 i 31.A 32. A In Exercises 45Ð48, use the result of Exercise 44 to determine a, b, 1 i 2 i 3 and c so that A is unitary. i i 2 3 4i 2 2 1 1 a 1 a 45.A 46. A 5 i 2 b c 2 33. A 20 b c 2 6 3i i 1 i a 1 a 02 47.A 48. A 45 2 2 b c 2 b c 1 4 1 i 34. A 0 i 3i In Exercises 49Ð52, prove the given formula, where A and B are n n complex matrices. 0 0 2 i 49.A A 50. A B A B In Exercises 35Ð38, determine the eigenvectors of the matrix. 51.kA kA 52. AB BA 35. The matrix in Exercise 29. 53. Let A be a matrix such that A A O. Prove that iA is 36. The matrix in Exercise 30. Hermitian. 37. The matrix in Exercise 33. 54. Show that detA detA, where A is a 2 2 matrix. 38. The matrix in Exercise 32. In Exercises 55Ð56, assume that the result of Exercise 54 is true for In Exercises 39Ð43, find a unitary matrix P that diagonalizes the matrices of any size. matrix A. 55. Show that detA detA. 0 i 0 2 i 56. Prove that if A is unitary, then detA 1. 39.A 40. A i 0 2 i 4 57. (a) Prove that every Hermitian matrix A can be written as the i i sum A B iC, where B is a real symmetric matrix and 2 C is real and skew-symmetric. 2 2 (b) Use part (a) to write the matrix i 33. A 20 2 2 1 i A i 1 i 3 02 2 as a sum A B iC, where B is a real symmetric matrix and C is real and skew-symmetric. 4 2 2i 42. A 2 2i 6 (c) Prove that every n n complex matrix A can be written as A B iC, where B and C are Hermitian. 1 0 0 43. A 0 1 1 i (d) Use part (c) to write the complex matrix 0 1 i 0 i 2 A 2 i 1 2i as a sum A B iC, where B and C are Hermitian. SECTION 8.5 EXERCISES 511

58. Determine which of the following sets are subspaces of the (d) Find a 2 2 matrix that is unitary, but not Hermitian. vector space of n n complex matrices. (e) Find a 2 2 matrix that is normal, but neither Hermitian (a) The set of n n Hermitian matrices. nor unitary. (b) The set of n n unitary matrices. (f) Find the eigenvalues and corresponding eigenvectors of (c) The set of n 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 2 matrix that is Hermitian, but not unitary. is not diagonalizable. Is this matrix normal?