Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.5

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.5

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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. Note that if A is a matrix with real entries, then A *. 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 A 5 3 3 1 7 i 2 i 0 4 2 i 4 . 5 A T Definition of the Conjugate Transpose of a Complex Matrix The conjugate transpose of a complex matrix A , denoted by A*, is given by A* where the entries of are the complex conjugates of the corresponding entries of A. A 5 A T
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Solution 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. Unitary Matrices Recall that a real matrix A is orthogonal if and only if In the complex system, matrices having the property that * are more useful and such matrices are called unitary. EXAMPLE 2 A Unitary Matrix Show that the matrix is unitary. Solution Because you can conclude that So, A is a unitary matrix. A * 5 A 2 1 . AA * 5 1 2 3 1 1 i 1 2 i 1 2 i 1 1 i 4 1 2 3 1 2 i 1 1 i 1 1 i 1 2 i 4 5 1 4 3 4 0 0 4 4 5 I 2 , A 5 1 2 3 1 1 i 1 2 i 1 2 i 1 1 1 4 A 2 1 5 A A 2 1 5 A T . A * 5 A T 5 3 3 2 7 i 0 2 2 i 4 1 i 4 A 5 3 3 1 7 i 2 i 0 4 2 i 4 5 3 3 2 7 i 2 2 i 0 4 1 i 4 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 501 Definition of a Unitary Matrix A complex matrix A is unitary if A 2 1 5 A *. Theorem 8.8 Properties of Conjugate Transpose If A and B are complex matrices and k is a complex number, then the following proper- ties are true. 1. 2. 3. 4. s AB d * 5 B * A * s kA d * 5 k A * s A 1 B d * 5 A * 1 B * s A * d * 5 A
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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 in (complex Euclidean space) is called orthonormal if the following are true. 1. 2. The proof of the following theorem is similar to the proof of Theorem 7.8 given in Section 7.3. 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 Solution Let and be defined as follows. The length of is 5 3 1 4 1 2 4 1 1 4 4 1 y 2 5 1. 5 3 1 1 2 21 1 2 2 1 1 1 1 i 2 1 1 i 2 2 1 1 2 1 2 2 1 2 2 4 1 y 2 i r 1 i 5 s r 1 ?
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.5

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