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* 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 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 ϩ 7i 0 A ϭ ΄ ΅. 2i 4 Ϫ i SECTION 8.5 UNITARY AND HERMITIAN MATRICES 501 Solution 3 ϩ 7i 0 3 Ϫ 7i 0 A ϭ ϭ ΄ ΅ ΄ 2i 4 Ϫ i΅ Ϫ2i 4 ϩ i 3 Ϫ 7i Ϫ2i A* ϭ AT ϭ ΄ ΅ 0 4 ϩ 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 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*͒* ϭ A Conjugate Transpose 2. ͑A ϩ B͒* ϭ A* ϩ B* 3. ͑kA͒* ϭ kA* 4. ͑AB͒* ϭ 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 such matrices are called unitary. Definition of a A complex matrix A is unitary if Unitary Matrix AϪ1 ϭ A*. EXAMPLE 2 A Unitary Matrix Show that the matrix is unitary. 1 1 ϩ i 1 Ϫ i A ϭ ΄ ΅ 2 1 Ϫ i 1 ϩ 1 Solution Because 1 1 ϩ i 1 Ϫ i 1 1 Ϫ i 1 ϩ i 1 4 0 AA* ϭ ΄ ΅ ΄ ΅ ϭ ΄ ΅ ϭ I , 2 1 Ϫ i 1 ϩ i 2 1 ϩ i 1 Ϫ i 4 0 4 2 you can conclude that A* ϭ AϪ1. 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 matri- ces 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. ʈ ʈ ϭ ϭ 1. vi 1, i 1, 2, . , m ϭ 2. vi и 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 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 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 Ί3 Ί3 Ί3 5i 3 ϩ i 4 ϩ 3i 2Ί15 2Ί15 2Ί15 Solution Let r1, r2, and r3 be defined as follows. 1 1 ϩ i 1 r ϭ ΂ , , Ϫ ΃ 1 2 2 2 i i 1 r ϭ ΂Ϫ , , ΃ 2 Ί3 Ί3 Ί3 5i 3 ϩ i 4 ϩ 3i r ϭ ΂ , , ΃ 3 2Ί15 2Ί15 2Ί15 The length of r1 is ʈ ʈ ϭ ͑ ͒1͞2 r1 r1 и r1 1 1 1 ϩ i 1 ϩ i Ϫ1 Ϫ1 1͞2 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 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 Ϫi 1 ϩ i i Ϫ1 1 r и r ϭ ΂ ΃΂ ΃ ϩ ΂ ΃΂ ΃ ϩ ΂ ΃΂ ΃ 1 2 2 Ί3 2 Ί3 2 Ί3 1 i 1 ϩ i Ϫi Ϫ1 1 ϭ ΂ ΃΂ ΃ ϩ ΂ ΃΂ ΃ ϩ ΂ ΃΂ ΃ 2 Ί3 2 Ί3 2 Ί3 i i 1 1 ϭ Ϫ ϩ Ϫ 2Ί3 2Ί3 2Ί3 2Ί3 ϭ 0. ϭ ϭ ͭ ͮ Similarly,r1 и r3 0 and r2 и r3 0. So, you can conclude that r1, r2, r3 is an ortho- normal 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 (1822–1901). Definition of a A square matrix A is Hermitian if Hermitian Matrix A ϭ A*. As with symmetric matrices, you can easily recognize Hermitian matrices by inspection. To see this, consider the 2 ϫ 2 matrix A. ϩ ϩ ϭ a1 a2i b1 b2i A ΄ ϩ ϩ ΅ c1 c2i d1 d2i The conjugate transpose of A has the form A* ϭ AT ϩ ϩ ϭ a1 a2i c1 c2i ΄ ϩ ϩ ΅ b1 b2i d1 d2i Ϫ Ϫ ϭ a1 a2i c1 c2i ΄ Ϫ Ϫ ΅. b1 b2i d1 d2i If A is Hermitian, then A ϭ A*. So, you can conclude that A must be of the form ϩ ϭ a1 b1 b2i A ΄ Ϫ ΅. b1 b2i d1 504 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 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 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 both sides of the equation Av ϭ ␭v are multiplied by the row vector v*, then ϭ ͑␭ ͒ ϭ ␭͑ ͒ ϭ ␭͑ 2 ϩ 2 ϩ 2 ϩ 2 ϩ . ϩ 2 ϩ 2͒ v*Av v* v v*v a1 b1 a2 b2 an bn . Furthermore, because ͑v*Av͒* ϭ v*A*͑v*͒* ϭ v*Av, SECTION 8.5 UNITARY AND HERMITIAN MATRICES 505 it follows that v*Av is a Hermitian 1 ϫ 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 ma- trices. EXAMPLE 5 Finding the Eigenvalues of a Hermitian Matrix Find the eigenvalues of the matrix A. 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 Խ ␭I Ϫ AԽ ϭ ΄Ϫ2 Ϫ i ␭ Ϫ1 ϩ i΅ Ϫ3i Ϫ1 Ϫ i ␭ ϭ ͑␭ Ϫ 3͒͑␭2 Ϫ 2͒ Ϫ ͑Ϫ2 ϩ i͓͒͑Ϫ2 Ϫ i͒␭ Ϫ ͑3i ϩ 3͔͒ ϩ 3i ͓͑1 ϩ 3i͒ ϩ 3␭i͔ ϭ ͑␭3 Ϫ 3␭2 Ϫ 2␭ ϩ 6͒ Ϫ ͑5␭ ϩ 9 ϩ 3i͒ ϩ ͑3i Ϫ 9 Ϫ 9␭͒ ϭ ␭3 Ϫ 3␭2 Ϫ 16␭ Ϫ 12 ϭ ͑␭ ϩ 1͒͑␭ Ϫ 6͒͑␭ ϩ 2͒. This implies that the eigenvalues of A are Ϫ1, 6, and Ϫ2. 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 ␭ ϭϪ1 is obtained by solving the following equation. ␭ Ϫ Ϫ ϩ 3 2 i 3i v1 0 Ϫ2 Ϫ i ␭ Ϫ1 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 Ϫ1 ϩ i v ϭ 0 ΄ ΅΄ 2΅ ΄ ΅ Ϫ Ϫ Ϫ Ϫ 3i 1 i 1 v3 0 506 CHAPTER 8 COMPLEX VECTOR SPACES Using Gauss-Jordan elimination, or a computer or calculator, obtain the following eigen- ␭ ϭϪ vector corresponding to 1 1.