Unitary Matrices and Hermitian Matrices

Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a − bi. The conjugate of a + bi is denoted a+ bi or (a+ bi)∗. In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3− 4i, 5− 6i = 5 + 6i, 7i = −7i, 10 = 10. Complex conjugation satisfies the following properties: (a) If z ∈ C, then z = z if and only if z is a real number. (b) If z1, z2 ∈ C, then z1 + z2 = z1 + z2. (c) If z1, z2 ∈ C, then z1 · z2 = z1 · z2. The proofs are easy; just write out the complex numbers (e.g. z1 = a+bi and z2 = c+di) and compute. The conjugate of a matrix A is the matrix A obtained by conjugating each element: That is,